Methods of constructing and designing RF pulses and exciting or inverting two-level systems

ABSTRACT

The present disclosure provides for a method of designing a radiofrequency or broadband pulse sequence. The method can comprise a qubit (e.g., nuclear spin, photon, electron, atomic spin, dot spin) and a harmonic oscillator wherein a flip angle is controlled by steering a spring between specific states.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims priority from U.S. Provisional Application Ser.No. 62/519,859 filed on 14 Jun. 2017, which is incorporated herein byreference in its entirety.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

This invention was made with government support under grant numberCMMI1462796 awarded by National Science Foundation. The government hascertain rights in the invention.

MATERIAL INCORPORATED-BY-REFERENCE

Not applicable.

FIELD OF THE INVENTION

The present disclosure generally relates to methods of generating pulsesequences.

SUMMARY OF THE INVENTION

Among the various aspects of the present disclosure is the provision ofa designing a broadband radiofrequency pulse sequence.

An aspect the present disclosure provides for a method of performingbroadband excitation or inversion of a two-level spin system.

In some embodiments, the method can include providing a two-level systemcomprising at least one qubit (spin) and at least one harmonicoscillator (spring); defining a radio frequency (RF) bandwidth for thetwo-level system; bounding a radio frequency (RF) amplitude or totalenergy for an RF pulse; determining a desired terminal magnetizationprofile or flip angle for the spins; determining a desired terminalmagnetization profile or flip angle for the springs; mapping the springsto spins comprising mapping an endpoint of a trajectory of the spins toan endpoint of a trajectory of the springs, wherein the endpoints of thespins and springs correspond to the desired terminal magnetizationprofile or flip angle; employing a single control or two controlssimultaneously in both an x direction and a y direction; calculating aconverging solution for an RF pulse; or steering the spring and the spinto a desired terminal magnetization profile or flip angle by applyingthe calculated converging solution for the RF pulse to the two-levelsystem.

In some embodiments, the calculating a converging solution comprisesgenerating RF pulse parameters, providing a first control designcomprising a minimum-energy broadband pulse, or providing a secondcontrol design comprising an amplitude-limited broadband pulse.

In some embodiments, the two-level spin system is selected from thegroup consisting of a logical qubit spin system, a nuclear spin system,a photon spin system, an electron spin system, an atomic spin system,and a dot spin system.

In some embodiments, a dynamic connection between nonlinear spin andlinear spring systems are calculated under optimal forcing to design anRF pulse based on the design of a control to steer linear harmonicoscillators In some embodiments, a condition of a one-to-onecorrespondence to a spin trajectory is satisfied.

In some embodiments, the RF pulse conditions result in the spins andsprings having coinciding magnetization on the same axis and excitationor inversion of at least 99% of the spins and springs.

In some embodiments, the RF pulse compensates for a distribution of spinand spring frequencies.

In some embodiments, the RF pulse is an excitation pulse, a reverseexcitation pulse; or an inversion pulse.

In some embodiments, the RF pulse results in a spin flip angle selectedfrom π, π/2, or π/4.

In some embodiments, the RF pulse is computed using linear systems toforce or steer the spring.

In some embodiments, the RF pulse is an excitation pulse or an inversionpulse for a nonlinear Bloch system.

Another aspect of the present disclosure provides for a method ofconstructing an RF pulse.

In some embodiments, the method includes obtaining an energy parameterobtaining a bandwidth parameter obtaining a desired flip angle(magnetization profile), wherein the flip angle is between 0 and 180degrees; converting the desired flip angle from a spatial coordinatesystem to a linear coordinate system; inputting the energy parameter,the bandwidth parameter, and the linear coordinates into the system; oranalytically deriving parameters for the RF pulse. In some embodiments,the RF pulse produces a desired magnetization profile of a final spin orspring state.

In some embodiments, the bandwidth parameter corresponds to a range offrequencies of a sample; the method can be performed on-line; or themethod uses analytical approaches or optimization-free algorithms.

In some embodiments, the method includes providing a two-level systemcomprising at least one qubit (spin) and at least one harmonicoscillator (spring) and mapping the spins to the springs; or providing amaximum allowable amplitude a bound on an RF pulse amplitude for use ina broadband RF pulse.

In some embodiments, the method includes providing a first controldesign comprising a minimum-energy broadband pulse; or providing asecond control design comprising an amplitude-limited broadband pulse

In some embodiments, the RF pulse produces a desired distribution offinal spin or spring states or a desired magnetization profile whenapplied to a two-level spin system; achieves at least 99% broadbandexcitation or inversion; or is not a hyperbolic secant pulse.

In some embodiments, the RF pulse compensates for a distribution of spinand spring frequencies; or satisfies experimental requirements selectedfrom a bound on the RF pulse amplitude or total energy of the RF pulse.

In some embodiments, the bandwidth parameter is between −40 kHz and 40kHz.

In some embodiments, the energy parameter comprises a maximum allowablepower in Watts or dbm.

In some embodiments, flip angle is between 0 and 360° or selected from aπ, a π/2, or a π/4 flip angle.

Yet another aspect of the present disclosure provides for a method ofdesigning a broadband radiofrequency (RF) pulse sequence.

In some embodiments the method includes providing a two-level spinsystem comprising at least one qubit (spin) and at least one harmonicoscillator (spring); employing at least a first control design andoptionally, a second control design simultaneously in both an x and a ydirection; or achieving a desired flip angle of a qubit. In someembodiments, the flip angle of a qubit is controlled by steering thespring between specific states.

In some embodiments, the qubit is selected from the group consisting ofa logical qubit, a nuclear spin, a photon, an electron, an atomic spin,and a dot spin. In some embodiments, the qubit is a nuclear spin and abroadband RF pulse compensates for a distribution of spin and springfrequencies.

In some embodiments, the method includes numerical optimization,resulting in an RF pulse or an RF pulse sequence that places bounds on aradio-frequency (RF) amplitude or a total energy of the RF pulse.

In some embodiments, the control comprises the first control designcomprises a minimum-energy broadband pulse; and the second controldesign comprises an amplitude-limited broadband pulse; the flip anglecan be between about 0° and 360°; or the RF pulse performs at least 99%,exact, or asymptotically exact excitation or inversion over a definedbandwidth, optionally with a bounded amplitude.

In some embodiments, the RF pulse has a bang-bang pulse shape.

In some embodiments, an excitation or inversion of the spins can beadjusted by selecting different amplitude bounds and RF pulse durations.

Other objects and features will be in part apparent and in part pointedout hereinafter.

DESCRIPTION OF THE DRAWINGS

Those of skill in the art will understand that the drawings, describedbelow, are for illustrative purposes only. The drawings are not intendedto limit the scope of the present teachings in any way.

FIG. 1. The minimum-energy control u*_(π/2) steering the spring fromX₀=(0, 0) to X_(π/2)=(π/2, 0), with the corresponding trajectories ofthe spring (red) and spin (black) for ω=3 and T=π.

FIG. 2. The (a) broadband minimum-energy and (b) amplitude-limitedcontrols steering a family of harmonic oscillators with frequencies−1≤ω≤1 from X₀=(0, 0) to X_(π/2)=(π/2, 0), with the correspondingtrajectories, (d) and (e), respectively, of the harmonic oscillator(red) and nuclear spin (black) for ω=−1. (c) These pulses achieveaverage magnetization excitation, M_(x)(T), over the frequencies −1≤ω≤1,of 1.000 and 0.996, respectively.

FIG. 3. The trajectory of |f|² and the corresponding bound given by thesquare root of the right side of equation (4) resulting from theminimum-energy control u*_(π/2) (top) and a quadratic control,u(t)=(18t²+4−9π²)/8, (bottom) that steer the spring from X₀=(0, 0) toX_(π/2)=(π/2, 0).

FIG. 4. From top to bottom, the minimum-energy control u*(t)=−cos(3t) asin (7) for ω=3 and T=π and the corresponding spin trajectory (x(t),y(t), z(t)) and the evolutions of the complex-valued functions a(t),f(t), and g(t) following u*(t). Note that the final state of a, a(T), islarge since m(T) approaches to 0.

FIG. 5. The minimum-energy control steering the spring from (0, 0)′ to(π, 0)′, with the corresponding trajectories of the spring (red) andspin (black) for ω=3 and T=π, where the final z magnetization of thespin is −0.9851. The final z magnetization can be made arbitrarily closeto −1 by selecting the product ωT appropriately (see Section 2.4 ofExample 2); for parameters ω=3 and T=10π, z(T)=−0.9999.

FIG. 6. The minimum energy control u*(t) evidences improving final zmagnetization as the product ωT increases. An alternative derivationprovides insight into why this is by revealing that the asymptoticbehavior of a(T) drives this performance improvement. For example, whenωT=30π, z(T)=0.99999.

FIG. 7. The minimum energy control u*(t)=−cos(3t) as in (7) for ω=3 andT=π, the corresponding spin (m(t), z(t)) and spring X(t) trajectories,and the evolutions of the complex-valued functions f(t), g(t), and a(t)following u*(t). Since the minimum energy control u*(t) satisfies (5),the final value of g is close to g(T)≈4/(4+π)≈0.56. The control alsodrives a(T)≈−1. The integral condition guarantees the spring is drivento zero and when u*(t) additionally satisfies a(T)→−1 the spin is drivento the +z pole.

FIG. 8. The combined minimum energy controls a(t)=u*(t)+iv*(t)=−¼cos(3t)−i¼ cos(3t)−i¼ cos(3t) executes a transfer of the spin fromspherical coordinates θ=π/3 (azimuthal angle), ϕ=π/4 (polar angle),i.e.,

${{M(0)} = \left( {\frac{1}{2\sqrt{2}},\frac{\sqrt{3}}{2\sqrt{2}},\frac{1}{\sqrt{2}}} \right)^{r}},$to θ=0, ϕ=0, i.e., M(T)=(0, 0, 1)′, for ω=3 and T=π. These controls aredesigned separately as described in (54) and (55) to drive the springfrom

${\overset{\sim}{m}(0)} = {\frac{\pi}{8} + {i\frac{\pi\sqrt{3}}{8}}}$to {tilde over (m)}(T)=0. We show the control pulse, the correspondingspin (m(t), z(t)) and spring X(t) trajectories, the evolutions of thecomplex-valued functions f(t), g(t), and a(t) following α(t), and thetrajectories visualized on the sphere. Notice that the values of f(0)and g(0) are set to accommodate the different initial starting state.The control achieves a transfer with performance z(t)=0.99998. Thisexample, and others like it, empirically demonstrates that the frameworkpresented here is able to generalize to both the setting of arbitraryrotation angles as well as the design of two controls to execute theserotations in arbitrary directions.

FIG. 9. The trajectory of the discriminant of D as given by (59)resulting from the minimum-energy control; a linear control, u(t)=9/4t−9/8 π²; a quadratic control u(t)=9/4 t²−9/8 π²+½; and an exponentialcontrol

${u(t)} = {{\frac{5\pi}{1 + e^{\pi}}e^{t}} - \frac{9}{4}}$that steer the spring from (π/2, 0)′ to (0, 0)′.

The performance (the z magnetization at T=π) following each control is0.9991, −0.7647, 0.2127, −0.4205, 0.9525, respectively (the performanceof the exponential control, 0.9525, is a coincidence at these parametervalues). Recall (see Section 2.4 of Example 2) that the performance ofthe minimum energy pulse can be made arbitrarily close to 1 (e.g.,increasing the duration to T=10π yields a performance z(T)=0.999991).The well-known on-resonance pulse, α(t)=−½e^(i3t), which uses twocontrols (u(t)=−½ cos 3t, v(t)=−½ sin 3t) to perform exact excitation ofa single spin (but is not extendable to the broadband case) is alsoshown as a comparison and also satisfies the discriminant condition in(59).

FIG. 10. The broadband minimum-energy control steers a family ofharmonic oscillators with frequencies −1≤ω≤1 from (0, 0)′ to (π, 0)′,with the corresponding trajectories of the harmonic oscillator (red) andnuclear spin (black) for ω=−1. This pulse achieves an averagemagnetization inversion, over the frequencies −1≤ω≤1 of −0.9984.

DETAILED DESCRIPTION OF THE INVENTION

The present disclosure is based, at least in part, on the discovery thata nontrivial dynamic connection between nonlinear spin and linear springsystems under optimal forcing. As disclosed herein, it was shown that aproperly conditioned forcing of the linear harmonic oscillator, whichcan be computed using known linear systems theory, is an excitationpulse for the nonlinear Bloch system. Disclosed herein is the surprisingresult that such nonlinear and complex pulse design problems areequivalent to designing controls to steer linear harmonic oscillators,under certain conditions. We derive analytic broadband π/2 and π pulsesthat perform exact, or asymptotically exact, excitation and inversionover a defined bandwidth, and also with bounded amplitude. As shownherein, the present disclosure provides for a framework that provides amethodology to design analytic broadband pulses for controlling quantumsystems. The present disclosure also provides for analytical quantumpulses that can be computed with minimal numerical computationalefforts, and will have significant impact on the areas of nuclearmagnetic resonance (NMR) spectroscopy and magnetic resonance imaging(MRI), quantum computation, and quantum information processing, becausethese applications are enabled by a single or a sequence of pulses. Assuch, the presently disclosed analytical pulse design protocols can beused on MRI scanners and NMR spectrometers to obtain high-resolutionmedical images and protein spectroscopy. The present disclosure is alsodescribed in Li et al. Nature Comm. 8:446.

The present disclosure reveals a dynamic connection between theevolution of nuclear spins and harmonic oscillators (springs) driven bythe same external control input. In particular, we are able to create90° and 180° pulses used commonly in nuclear magnetic resonance (NMR) bydesigning controls that steer a spring between specific states. Thelinearity of the spring dynamic enables us to obtain simple analyticexpressions for these NMR pulses. We extend this insight to spin andspring ensembles in order to design broadband NMR pulses that compensatefor a distribution of spin (and spring) frequencies. Using thistechnique in conjunction with efficient numerical optimization we candesign pulse sequences that also satisfy specific experimentalrequirements, such as a bound on the radio-frequency (RF) amplitude orthe total energy of the pulse. Frequency can be in s⁻¹ (Hz), radians,etc.

As described herein, the RF pulse can also be a sequence of RF pulses toprovide a desired spin flip angle.

Designing accurate and high-fidelity broadband pulses is an essentialcomponent in conducting quantum experiments across fields from proteinspectroscopy to quantum optics. However, constructing exact and analyticbroadband pulses remains unsolved due to the nonlinearity and complexityof the underlying spin system dynamics. We reveal a nontrivial dynamicconnection between nonlinear spin and linear spring systems underoptimal forcing and present the surprising result that such nonlinearand complex pulse design problems are equivalent to designing controlsto steer linear harmonic oscillators, under certain conditions. Wederive analytic broadband π/2 and π pulses that perform exact, orasymptotically exact, excitation and inversion over a defined bandwidth,and also with bounded amplitude.

The presently disclosed methods provide for an advancement over thestate of the art. For example, current methods for pulse optimizationare based on numerical optimization with software using large scale,non-linear approaches. Such approaches may or may not converge and aretime intensive. Such approaches cannot be currently performed on-line.The methods described herein converge and can be performed on-line.

The methods of the present disclosure provide other advantages overcurrent methods. For example, as described herein are methods ofdesigning an RF pulse with precise fidelity (100% or asymptoticallyclose to 100%) for an entire bandwidth of qubits (e.g., nuclear spins)given experimental parameters (e.g., power constraints and desiredbandwidth). The RF pulse is designed by converting a non-linear designproblem into a linear design problem. Advantages of the disclosedmethods include (i) the application can be performed in situ, on-line,in real-time, with no need for re-computing; (ii) the solution alwaysconverges (because the non-linear problem is converted into a linearproblem); (iii) the designed pulse excites the entire spin system,allowing for an increased spin magnetization/alignment (e.g., especiallyfor “off-resonance” spins), resulting in increased SNR, increased phasecoherence, and improved resolution; and (iv) can use lower or sameenergy and obtain same or better results, respectively.

Applications for the disclosed methods include, but are not limited to,controlling spin systems, including NMR (e.g., protein NMR), MRI, EPR,ESR, SNR, ultra-fast optical devices, quantum computing, quantum gating,and quantum information processing.

Spin is a form of angular momentum carried by elementary particles. Itwas discovered that the solution can be solved analytically because spinsystems are nonlinear and difficult to work with, while spring systems,or harmonic oscillators, are linear and easier to work with.

Spin dynamics, pulse sequences and other NMR pulse sequence designprocesses are well known; see e.g., Spin Dynamics: Basics of NuclearMagnetic Resonance, Edition 2 Malcolm H. Levitt May 20, 2013 John Wiley& Sons; Understanding NMR Spectroscopy: Edition 2 James Keeler Sep. 19,2011 John Wiley & Sons. Except as otherwise noted herein, therefore, theprocess of the present disclosure can be carried out in accordance withsuch processes.

Definitions and methods described herein are provided to better definethe present disclosure and to guide those of ordinary skill in the artin the practice of the present disclosure. Unless otherwise noted, termsare to be understood according to conventional usage by those ofordinary skill in the relevant art.

In some embodiments, numbers expressing quantities of ingredients,properties such as molecular weight, reaction conditions, and so forth,used to describe and claim certain embodiments of the present disclosureare to be understood as being modified in some instances by the term“about.” In some embodiments, the term “about” is used to indicate thata value includes the standard deviation of the mean for the device ormethod being employed to determine the value. In some embodiments, thenumerical parameters set forth in the written description and attachedclaims are approximations that can vary depending upon the desiredproperties sought to be obtained by a particular embodiment. In someembodiments, the numerical parameters should be construed in light ofthe number of reported significant digits and by applying ordinaryrounding techniques. Notwithstanding that the numerical ranges andparameters setting forth the broad scope of some embodiments of thepresent disclosure are approximations, the numerical values set forth inthe specific examples are reported as precisely as practicable. Thenumerical values presented in some embodiments of the present disclosuremay contain certain errors necessarily resulting from the standarddeviation found in their respective testing measurements. The recitationof ranges of values herein is merely intended to serve as a shorthandmethod of referring individually to each separate value falling withinthe range. Unless otherwise indicated herein, each individual value isincorporated into the specification as if it were individually recitedherein.

In some embodiments, the terms “a” and “an” and “the” and similarreferences used in the context of describing a particular embodiment(especially in the context of certain of the following claims) can beconstrued to cover both the singular and the plural, unless specificallynoted otherwise. In some embodiments, the term “or” as used herein,including the claims, is used to mean “and/or” unless explicitlyindicated to refer to alternatives only or the alternatives are mutuallyexclusive.

The terms “comprise,” “have” and “include” are open-ended linking verbs.Any forms or tenses of one or more of these verbs, such as “comprises,”“comprising,” “has,” “having,” “includes” and “including,” are alsoopen-ended. For example, any method that “comprises,” “has” or“includes” one or more steps is not limited to possessing only those oneor more steps and can also cover other unlisted steps. Similarly, anycomposition or device that “comprises,” “has” or “includes” one or morefeatures is not limited to possessing only those one or more featuresand can cover other unlisted features.

All methods described herein can be performed in any suitable orderunless otherwise indicated herein or otherwise clearly contradicted bycontext. The use of any and all examples, or exemplary language (e.g.“such as”) provided with respect to certain embodiments herein isintended merely to better illuminate the present disclosure and does notpose a limitation on the scope of the present disclosure otherwiseclaimed. No language in the specification should be construed asindicating any non-claimed element essential to the practice of thepresent disclosure.

Groupings of alternative elements or embodiments of the presentdisclosure disclosed herein are not to be construed as limitations. Eachgroup member can be referred to and claimed individually or in anycombination with other members of the group or other elements foundherein. One or more members of a group can be included in, or deletedfrom, a group for reasons of convenience or patentability. When any suchinclusion or deletion occurs, the specification is herein deemed tocontain the group as modified thus fulfilling the written description ofall Markush groups used in the appended claims.

All publications, patents, patent applications, and other referencescited in this application are incorporated herein by reference in theirentirety for all purposes to the same extent as if each individualpublication, patent, patent application or other reference wasspecifically and individually indicated to be incorporated by referencein its entirety for all purposes. Citation of a reference herein shallnot be construed as an admission that such is prior art to the presentdisclosure.

Having described the present disclosure in detail, it will be apparentthat modifications, variations, and equivalent embodiments are possiblewithout departing the scope of the present disclosure defined in theappended claims. Furthermore, it should be appreciated that all examplesin the present disclosure are provided as non-limiting examples.

EXAMPLES

The following non-limiting examples are provided to further illustratethe present disclosure. It should be appreciated by those of skill inthe art that the techniques disclosed in the examples that followrepresent approaches the inventors have found function well in thepractice of the present disclosure, and thus can be considered toconstitute examples of modes for its practice. However, those of skillin the art should, in light of the present disclosure, appreciate thatmany changes can be made in the specific embodiments that are disclosedand still obtain a like or similar result without departing from thespirit and scope of the present disclosure.

Example 1: Exact Broadband Excitation of Two-Level Systems—Mapping Spinsto Springs

This example describes the derivation of analytic broadband pulses(e.g., π/2 and π pulses) that perform exact, or asymptotically exact,excitation and inversion over a defined bandwidth, and also with boundedamplitude.

Manipulating large ensembles of dynamical systems by a single controlsignal is a common challenging problem in experimental physics,chemistry, biology, and neuroscience; difficult chiefly due to theinherent heterogeneity present in the systems. For example, inspectroscopic applications such as nuclear magnetic resonance (NMR)spectroscopy, optical spectroscopy, magnetic resonance imaging (MRI),and quantum computing, experiments are often performed on quantumensembles on the order of 10²³ rather than individual molecules or atoms(1, 2, 3, 4, 5, 6, 7). These experiments are conducted through asequence of engineered electromagnetic pulses that produce a desiredexcitation profile or a time evolution of the quantum ensemble. NMRspectroscopy offers canonical examples, including broadband excitationand inversion of spin populations, where a single coil is available togenerate radio-frequency (RF) fields that are used to steer all nuclearspins of a particular type (e.g., protons) in a macroscopic sample fromsome initial state (e.g., thermal equilibrium) to a desired target state(e.g., transverse magnetization with a desired phase). The ability toperform a state-to-state transformation of this type has direct impacton experimental outcomes in, for example, increasing medical imageresolution (8), amplifying off-resonance signal recovery inlarge-protein NMR (9), achieving effective coherent control of logicalqubits (10), and developing pulses for ultrafast all-optical signalprocessing devices (11).

The Bloch model is a semi-classical description that describes theevolution of a two-level system and is of the form,

$\begin{matrix}{{{\frac{d}{dt}\begin{bmatrix}{M_{x}\left( {t,\omega} \right)} \\{M_{y}\left( {t,\omega} \right)} \\{M_{z}\left( {t,\omega} \right)}\end{bmatrix}} = {\begin{bmatrix}0 & {- \omega} & {u(t)} \\\omega & 0 & {- {v(t)}} \\{- {u(t)}} & {v(t)} & 0\end{bmatrix}\begin{bmatrix}{M_{x}\left( {t,\omega} \right)} \\{M_{y}\left( {t,\omega} \right)} \\{M_{z}\left( {t,\omega} \right)}\end{bmatrix}}},} & (1)\end{matrix}$

when the duration of the external RF pulse, T, is much shorter than thetransverse and longitudinal relaxation times T₂ and T₁ (1). In thiscase, the effects of relaxation can be neglected and the bulk “spin”magnetization vector M=(M_(x), M_(y), M_(z)) evolves on a sphere. In theabsence of an irradiating RF pulse for a time that is much longer thanT₁, the spin vector aligns with the static magnetic field,conventionally in the +z direction (the spin magnitude is typicallynormalized so we consider a unit sphere). To simplify the presentationof the analysis, we consider a pulse applied only along the y-axis,i.e., u(t)=−γB_(1y) and v(t)=−γB_(1x)=0, where B₁ is the strength of theapplied RF pulse and γ is the gyromagnetic ratio (we discuss animplementation with both controls in Section 2.5 of Example 2).

If the spin population is characterized by a single frequency, it iswell known that an on-resonance sinusoidal pulse, where the frequency ofthe RF pulse is tuned to match the frequency of the sample, is able toexcite or invert the spins exactly. In these cases, equalization (a π/2or 90° rotation) or inversion (a π or 180° rotation) of the spinpopulation can be achieved in T=π/(2γB₁) or T=π/(γB₁) units of time,respectively (12). For the sake of simplicity, the rest of themanuscript uses dimensionless variables, normalized by the value of γB₁.

In practice, however, the resonance frequencies of the spins are spreadover a range due to chemical shifts caused by varying levels of magneticshielding (1) or by magnetic field gradients (8). Practitionersregularly use this frequency dispersion in constructive ways in order todistinguish between nuclei in different chemical environments, however,the same phenomenon makes manipulating spin populations uniformly over aspecified bandwidth highly nontrivial.

Calculating the time evolution of the spin magnetization correspondingto a given RF pulse can be accomplished through straightforwardintegration, however, solving the “inverse” pulse design problem, whichseeks to construct an RF pulse that produces a desired distribution offinal spin states M(T), or magnetization profile, is much moredifficult. Work to date has focused on developing robust numericaloptimization techniques to search for an optimal pulse that achievesbroadband excitation or inversion. These methods are often highlycustomized and have slow or unverified convergence rates, especiallywhen designing pulse sequences for difficult experiments with moredemanding performance specifications (13). The development of analyticalapproaches or optimization-free algorithms for broadband pulse designhas been minimal due to the nonlinearity of the spin dynamics. Theexceptions are the hyperbolic secant pulse (14), which is aparameter-dependent selective inversion pulse, where the selectivity isachieved when the amplitude of the pulse reaches above a threshold andwhen the pulse parameters are appropriately tuned, and the Shinnar-LeRoux algorithm, which maps the problem of selective pulse design to thedesign of finite impulse response (FIR) filters (15). In this work, wepresent an analytic result for broadband pulse design.

Results

Mapping Spins to Springs.

Consider separately the dynamics of an undamped harmonic oscillatorrepresented in matrix form,

$\begin{matrix}{{{\frac{d}{dt}\begin{bmatrix}{x\left( {t,\omega} \right)} \\{y\left( {t,\omega} \right)}\end{bmatrix}} = {{\begin{bmatrix}0 & {- \omega} \\\omega & 0\end{bmatrix}\begin{bmatrix}{x\left( {t,\omega} \right)} \\{y\left( {t,\omega} \right)}\end{bmatrix}} + {\begin{bmatrix}1 \\0\end{bmatrix}{u(t)}}}},} & (2)\end{matrix}$

where the state X=(x,y), ω, and u(t) represent the oscillator's velocityand position, frequency, and external forcing, respectively (see Section1 in Example 2). Observe that the unforced dynamics, u(t)=v(t)=0 inequation (1), of the transverse components (M_(x) and M_(y)) of the spinmagnetization coincide with the dynamics of an unforced, u(t)=0,undamped harmonic oscillator, or “spring”, of the same frequency. It isintriguing then to explore the possibility that the connection betweenthe spin and the spring can be preserved even when driven by a commonexternal input. In this report, we identify and characterize theunexpected dynamic connection between the time-evolution of forced spinsand springs that is not limited to the linear regime of small rotationangles. We exploit this connection to offer an analytic solution to thebroadband pulse design problem. Ultimately, there can be no directmapping for every point along the evolution of these linear andnonlinear systems; however, we discover a dynamic projection that mapsthe endpoints of the trajectory of a spin to that of a spring, which issufficient, because the pulse design problem is defined by the desiredterminal magnetization profile.

To develop this connection, we construct the complex projection,

$\begin{matrix}{{{f(t)} = \frac{{M_{x}(t)} + {{iM}_{y}(t)}}{{a(t)} + {M_{z}(t)}}},} & (3)\end{matrix}$

where 0≤t≤T is the pulse duration and a(t)=a₁(t)+ia₂(t) is acomplex-valued function satisfying a Riccati equation with the initialcondition a(0)=1 and depending on the time-varying RF pulse (seeequation (13) in Section 2.1 of Example 2). If a(t)=1 over the entireduration, then f(t) simply becomes the stereographical projection. Usingthe fact that the magnitude of the vector ∥M∥=1, we can composeconditions on f(t) and a(t) to ensure that the dynamic projection yieldsa valid (i.e., noncomplex-valued and unique) Bloch trajectory (seeSection 3 of Example 2). The necessary and sufficient condition for theprojection in equation (3) to be a one-to-one correspondence to a spintrajectory is in terms of the following bound on f(t),

$\begin{matrix}{{0 \leq {f}^{2} < \frac{1 - {a}^{2} + \sqrt{\left( {1 - {a}^{2}} \right)^{2} + {4a_{2}^{2}}}}{2a_{2}^{2}}},} & (4)\end{matrix}$

where |a|²=a₁ ²+a₂ ². This condition also indicates why thestereographic projection fails to provide a mapping in the general case(see Section 3 of Example 2).

Using this dynamic projection, we show that an RF pulse, u(t), whichresults in f(t) that satisfies equation (4) and the following integralcondition,

$\begin{matrix}{{{\int_{0}^{T}{{u(t)}e^{i\;\omega\; t}{dt}}} = {- \frac{\pi}{2}}},} & (5)\end{matrix}$

steers the Bloch system from M(0)=(1, 0, 0) to M(T)=(0, 0, 1)−a reverseexcitation pulse (see Section 2.1 of Example 2). Classic linear systemsanalysis (16) reveals that the external forcing that steers a harmonicoscillator of the same frequency from X(0)=(π/2, 0) to X(T)=(0, 0) mustsatisfy the same integral condition in equation (5), therefore, underthe bound in equation (4), the conditions on the controls for drivingthe spin and spring coincide. More precisely, we show that u(t)satisfying equation (4) will exactly transfer the value of the dynamicprojection f(t) from 1 to 0, and hence drives a reverse excitation ofthe spin in the Bloch system, in the absence of irregular singularitiescaused by the evolution of a(t) at time T (see Sections 2.2-2.4 ofExample 2). The input that steers the spring from X₀=(0, 0) toX_(π/2)=(π/2, 0) at time T can be converted to a forward excitationpulse taking the spin from M₀=(0, 0, 1) to M_(π/2)=(1, 0, 0) byreversing it in time and changing its sign (17).

Analytical Optimal Excitation Pulses.

Most importantly—and the fundamental result reported in this work—aproperly conditioned forcing of the linear harmonic oscillator, whichcan be computed using known linear systems theory, is an excitationpulse for the nonlinear Bloch system. Among the many potential controlfunctions u(t) that complete the desired transfer, we can select theminimum-energy control, i.e., the control u*(t) that minimizes the costfunctional ∫₀ ^(T) u²(t)dt (see Section 1 of Example 2). For example,the minimum-energy control that steers the spring with frequency ω=3from X₀ to X_(π/2) at T=π is given by u*_(π/2)(t)=−cos(3t), which is aπ/2 pulse taking the spin from M₀ to M_(π/2). This optimal control andthe resulting trajectories of the spring and the spin with ω=3 areillustrated in FIG. 1.

The same notion can be adopted to design an inversion pulse, which isrealized by constructing a control that steers the spring from X₀ toX(T)=(π,0)=X_(π), or by concatenating a π/2 pulse with its time-reversedversion (17), which is a pulse sequence X₀ to X_(π/2) to X_(π). Theminimum-energy inversion pulse and the resulting trajectories for thespring and the spin of ω=3 and T=π are illustrated in FIG. 5 in Section2.1 of Example 2.

Broadband Excitation Pulses.

The dynamic connection between spin and spring has enabled the analyticdesign of π/2 and π pulses that manipulate the spin magnetization at asingle frequency ω. We now apply this discovery to design a control u(t)that simultaneously steers an ensemble of springs between X₀ and X_(π/2)(or X_(π)), which is called a broadband π/2 (or π) pulse, respectively.The minimum-energy broadband controls can be derived by solving theintegral equation (5) in function space (since ω becomes a variable) andare composed of the prolate spheroidal wave functions (see Section 4.1of Example 2). FIG. 2 and FIG. 10 show broadband π/2 and π pulses,respectively, which produce uniform excitation over the designedbandwidth. In practice these pulses can be constructed using thediscrete prolate spheroidal sequences available in many scientificprogramming tools, such as “dpss” in Matlab (see Section 4.2).

Practical considerations, e.g., limited power of RF coils, make itcritical to design pulses with bounded amplitude. Steering an ensembleof springs with a bounded control is a challenging optimal controlproblem. However, we show that it can be reduced to a convexoptimization problem, which can be effectively solved, and the optimalcontrol has a bang-bang pulse shape (see Section 4.2.1 of Example 2).The bang-bang pulse in FIG. 2 is an example of a bounded amplitudebroadband π/2 pulse. The performance (i.e., average excitation) can beadjusted by selecting different amplitude bounds and pulse durations.

DISCUSSION

The dynamic mapping in equation (3) reveals a nontrivial connectionbetween nonlinear spin and linear spring systems under optimal forcingand enables the design of analytical broadband pulses. The bound on thisdynamic projection f(t) in equation (4) is critical to ensure thefeasibility of the designed pulses. To illustrate the importance of thisbound, in FIG. 3 we plot the time evolution of |f(t)| for theminimum-energy control and a quadratic control, which satisfies theintegral condition in equation (5), but not the bound in equation (4).FIG. 9 in Section 3 shows several other such counterexample controls.Note that all controls steer the spring from X₀ to X_(π/2), but onlywhen equation (4) is satisfied in the minimum-energy case, does thecontrol also steer the spin from M₀ to M_(π/2).

Moreover, empirical results presented in Section 2.6 and FIG. 8)strongly suggest that the framework described here can be generalized todesign pulses that employ two controls simultaneously and also thatachieve arbitrary flip angles (not only restricted to π/2 and π pulses).

This work to derive conditions for analytic control inputs of spinsystems as well as the extension to exact broadband excitation andinversion opens up new avenues for pulse sequence design in quantumcontrol. The notion of dynamic projection methods on the spin-springrelationship introduced in the report, together with convex optimizationformulation to solve the optimal control problem with an amplitudebound, lays a foundation to develop pulse sequences for more complicatedprofiles, such as frequency selective pulses.

Methods

Optimal Steering of Springs.

The minimum-energy control that steers the spring modeled in equation(2) from X₀ to X_(F) can be derived using least squares theory (16) andis of the form u*(t)=B′e^(−At)W⁻¹[e^(−AT)X_(F)−X₀], where W is thecontrollability Gramian of the spring system, defined by W=∫₀^(T)e^(−At)BB′e^(−A′t)dt, where

$A = {{\begin{bmatrix}0 & {- \omega} \\\omega & 0\end{bmatrix}\mspace{14mu}{and}\mspace{14mu} B} = {\begin{bmatrix}1 \\0\end{bmatrix}.}}$For example, if the frequency of the spring is ω=3, then theminimum-energy control that drives the spring from X₀=(π/2,0) toX_(F)=(0, 0) of during π is u*(t)=−cos(3t) for t∈[0,π] (see Section 1).

Dynamic Mapping Between Spin and Spring.

Using the relation M_(x) ²+M_(y) ²+M_(z) ²=1 for all t∈[0, T], thecomplex projection defined in equation (3) follows the dynamic equation{dot over (f)}=iωf+½uf ²+½βu,f(0)=1,  (6)

where β=e^(2iωt), if the complex function α(t) is chosen to satisfy

${\overset{.}{a} = {{{- \frac{u\;\beta}{2m}}a^{2}} - {\frac{{uz}\left( {\beta - 1} \right)}{m}a} + \frac{u\left( {1 + z^{2} - {z^{2}\beta}} \right)}{2m}}},{{a(0)} = 1.}$

Integrating (6) using contour integration, we show that f is steered tof(T)=0 following the control input that drives the spring fromX₀=(π/2,0) to X_(F)=(0, 0). This implies that the spin is excited fromM₀=(0, 0, 1) to M_(π/2)=(1, 0, 0), so that this control is a π/2 (or90°) pulse (see Section 2).

REFERENCES

-   1. R. R. Ernst, G. Bodenhausen and A. Wokaun, Principles of Nuclear    Magnetic Resonance in Once and Two Dimensions. Clarendon Press,    Oxford (1987).-   2. D. Cory, A. Fahmy and T. Havel, Ensemble Quantum Computing by NMR    Spectroscopy, PNAS, 94, 1634-1639 (1997).-   3. J.-S. Li and N. Khaneja, Control of Inhomogeneous Quantum    Ensembles, Physical Review A, 73:030302 (2006).-   4. S. J. Glaser, et. al., Unitary Control in Quantum Ensembles,    Maximizing Signal Intensity in Coherent Spectroscopy, Science, 280,    421-424 (1998).-   5. B. Pryor and N. Khaneja, Fourier Decompositions and Pulse    Sequence Design Algorithms for Nuclear Magnetic Resonance in    Inhomogeneous Fields, Journal of Chemical Physics, 125:194111    (2006).-   6. M. S. Silver, R. I. Joseph, and D. I. Hoult, Selective Spin    Inversion in Nuclear Magnetic Resonance and Coherent Optics through    an Exact Solution of the Bloch-Riccati Equation, Physical Review A,    31(4), 2753-2755 (1985).-   7. N. Khaneja, J.-S. Li, C. Kehlet, B. Luy, and S. J. Glaser,    Broadband Relaxation-optimized Polarization Transfer in Magnetic    Resonance, Proceedings of the National Academy of Sciences, 101,    14742-14747 (2004).-   8. M. A. Bernstein, K. F. King, and X. J. Zhou, Handbook of MRI    Pulse Sequences (Elsevier Academic Press, 2004).-   9. D. P. Frueh, T. Ito, J.-S. Li, G. Wagner, S. J. Glaser and N.    Khaneja, Sensitivity Enhancement in NMR of Macromolecules by    Application of Optimal Control Theory, Journal of Biomolecular NMR    32, 23-30 (2005).-   10. J. R. Petta, et al., Coherent Manipulation of Coupled Electron    Spins in Semiconductor Quantum Dots. Science, 309, 2180 (2005).-   11. M. Santagiustina, S. Chin, N. Primerov, L. Ursini, and L.    Thavenaz, All-optical Signal Processing using Dynamic Brillouin    Gratings, Scientific Report, 3, 1594 (2013).-   12. J. Cavanagh, W. J. Fairbrother, A. G. Palmer, and N. J. Skelton,    Protein NMR Spectroscopy, Academic Press, San Diego, Calif. (1996).-   13. K. Kobzar, B. Luy, N. Khaneja, and S. J. Glaser, Pattern Pulses:    Design of Arbitrary Excitation Profiles as a Function of Pulse    Amplitude and Offset, Journal of Magnetic Resonance, 173, 229-235    (2005).-   14. M. S. Silver, R. I. Joseph, C.-N. Chen, V. J. Sank, and D. I.    Hoult, Selective population inversion in NMR, Nature, Vol. 310, 23    (1984).-   15. M. Shinnar and J. Leigh, The application of spinors to pulse    synthesis and analysis, Magnetic Resonance Med. 12, 93-98 (1989).-   16. R. W. Brocket, Finite Dimensional Linear Systems (Wiley, 1970).-   17. M. Braun and S. J. Glaser, Concurrently Optimized Cooperative    Pulses in Robust Quantum Control: Application to Broadband    Ramsey-Type Pulse Sequence Elements, New Journal of Physics 16,    115002 (2014).

Example 2: Supplemental Information

The major contribution of Example 1 was to reveal a dynamic connectionbetween the evolution of nuclear spins and harmonic oscillators(springs) driven by the same external control input. In particular, weare able to create 90° and 180° pulses used commonly in nuclear magneticresonance (NMR) by designing controls that steer a spring betweenspecific states. The linearity of the spring dynamic enables us toobtain simple analytic expressions for these NMR pulses. We extend thisinsight to spin and spring ensembles in order to design broadband NMRpulses that compensate for a distribution of spin (and spring)frequencies. Using this technique in conjunction with efficientnumerical optimization we can design pulse sequences that also satisfyspecific experimental requirements, such as a bound on theradio-frequency (RF) amplitude or the total energy of the pulse.

Section 1: Optimal Steering of Springs

The well-known simple harmonic oscillator obeys a dynamic described by{umlaut over ({tilde over (y)})}+ω²{tilde over (y)}=ũ, with {tilde over(y)} the position of the oscillator (or spring) and ũ is a forcing term.If we let

${\overset{\sim}{x} = {\frac{1}{\omega}\overset{.}{\overset{\sim}{y}}}},$then {dot over ({tilde over (y)})}=ω{tilde over (x)} and

${\overset{.}{\overset{\sim}{x}} = {{\frac{1}{\omega}\overset{¨}{\overset{\sim}{y}}} = {{{- \omega}\;\overset{\sim}{y}} + u}}},{{{where}\mspace{14mu} u} = {\frac{\overset{\sim}{u}}{\omega}.}}$Therefore, a forced, nondamped harmonic oscillator can be modeled as alinear dynamical system of the form,

$\begin{matrix}{{{\frac{d}{dt}{X(t)}} = {{{AX}(t)} + {{Bu}(t)}}},} & (1)\end{matrix}$

where X=({tilde over (x)}(t), {tilde over (y)}(t))′ represents the stateand ′ denotes the transpose operation,

$\begin{matrix}{{A = \begin{bmatrix}0 & {- \omega} \\\omega & 0\end{bmatrix}},\mspace{14mu}{B = \begin{bmatrix}1 \\0\end{bmatrix}},} & (2)\end{matrix}$in which ω is the frequency of the harmonic oscillator, and u: [0, T]→R,T∈(0, ∞), is an external input (control), which is (piecewise)continuous on [0, T]. We consider steering this harmonic oscillator fromthe initial state

$X_{0} = {\left( {{\overset{\sim}{x}(0)},{\overset{\sim}{y}(0)}} \right)^{\prime} = \left( {\frac{\pi}{2},0} \right)^{\prime}}$to the origin, X_(F)=(0, 0)′, at a finite time T. Applying the variationof constants formula [1] to (1) yieldsX(T)=e ^(AT) X ₀+∫₀ ^(T) e ^(A(T−σ)) Bu(σ)dσ.

Hence for X(T)=X_(F)=(0, 0)′, it requires that the control u(t)satisfies

$\begin{matrix}{{{\int_{0}^{T}{{u(\sigma)}{\cos({\omega\sigma})}d\;\sigma}} = {- \frac{\pi}{2}}},} & (3) \\{{{\int_{0}^{T}{{u(\sigma)}{\sin({\omega\sigma})}d\;\sigma}} = 0},} & (4)\end{matrix}$

or equivalently

$\begin{matrix}{{{\int_{0}^{T}{{u(t)}e^{i\;\omega\; t}{dt}}} = {- \frac{\pi}{2}}},} & (5)\end{matrix}$

in order to complete the desired transfer.

We know from linear systems theory [1] that the system (1) iscontrollable if ω≠0, and therefore there exists at least one controlu(t) that will accomplish the transfer from X₀ to X_(F). It is alsowell-known that the minimum-energy control, u*(t), that achieves thedesired transfer while minimizing the total energy, i.e., ∫₀^(T)u′(t)u(t)dt, takes the formu*(t)=B′e ^(−At) W ⁻¹ξ,  (6)

where ξ=e^(−AT)X_(F)−X₀ and W is the so-called controllability Gramian,given by

$W = {{\int_{0}^{T}{e^{- {At}}{BB}^{\prime}e^{{- A^{\prime}}t}{dt}}} = {\begin{bmatrix}{\frac{T}{2} + \frac{\sin\left( {2\omega^{\prime}T} \right)}{4\omega}} & {- \frac{\sin^{2}\left( {\omega\; T} \right)}{2\omega}} \\{- \frac{\sin^{2}\left( {\omega\; T} \right)}{2\omega}} & {\frac{T}{2} - \frac{\sin\left( {2\omega\; T} \right)}{4\omega}}\end{bmatrix}.}}$

For example, if the frequency of the spring is ω=3, then theminimum-energy control of duration π is

$\begin{matrix}{{{u^{*}(t)} = {\frac{3{\pi\left\lbrack {{\sin\left( {3\; t} \right)} + {\sin\left( {{6\; T} - {3\; t}} \right)} - {6\; T\;{\cos\left( {3\; t} \right)}}} \right\rbrack}}{\cos\left( {{6\; T} + {18\; T^{2}} - 1} \right.} = {- {\cos\left( {3\; t} \right)}}}},} & (7)\end{matrix}$

for t∈[0,π].

Section 2: Mapping Spins into Springs

The evolution of the bulk magnetization of a sample of nuclear spinsimmersed in an external magnetic field follows the Bloch equations [2],given by

$\begin{matrix}{{{\frac{d}{dt}\begin{bmatrix}{x(t)} \\{y(t)} \\{z(t)}\end{bmatrix}} = {\begin{bmatrix}0 & {- \omega} & {u(t)} \\\omega & 0 & {- {v(t)}} \\{- {u(t)}} & {v(t)} & 0\end{bmatrix}\begin{bmatrix}{x(t)} \\{y(t)} \\{z(t)}\end{bmatrix}}},} & (8)\end{matrix}$

where M(t)≐(x(t), y(t), z(t))′ is the magnetization vector andu=−γB_(1y) and v=−γB_(1x) are the applied RF fields in the y and x axis,respectively. As before, let m(t)=x(t)+iy(t) be the complex transversemagnetization and α(t)=u(t)−iυ(t) (the complex conjugate of α) be theirradiating RF field the Bloch equations may then be written in thecomplex form, that is,

where α denotes the complex conjugate of α.

Section 2.1 Dynamic Mapping Between Spin and Spring

Defining f: [0, T]→C by

$\begin{matrix}{{{f(t)} = \frac{m(t)}{{a(t)} + {z(t)}}},} & (11)\end{matrix}$

where a(t) is a complex-valued function over [0, T], then (9) and (10)can be transformed into the following Riccati equation,

$\begin{matrix}{{\overset{.}{f} = {{i\;\omega\; f} + {\frac{1}{2}\alpha\; f^{2}} + {\frac{1}{2}\overset{\_}{\alpha}} + \frac{{\overset{\_}{\alpha}\left( {1 - a^{2}} \right)} - {2\; m\;\overset{.}{a}}}{2\left( {a + z} \right)^{2}}}},} & (12)\end{matrix}$

in which we used mm+z²=1 since M (t) is conventionally a unit vector.Without loss of generality, we consider driving the Bloch equations asin (8) with one control letting v=0 (in later sections we consider thecase with two controls), and consider the canonical state transfer,equivalent to a 90° pulse, from the initial state M (0)=(1, 0, 0)′ inthe transverse plane (i.e., m(0)=1 and z(0)=0) to a final state M(T)=(0,0, 1)′ (i.e., m(T)=0 and z(T)=1).

Thus, we have α=α=u, and (12) becomes

$\overset{.}{f} = {{i\;\omega\; f} + {\frac{1}{2}{uf}^{2}} + {\frac{1}{2}u} + {\frac{{u\left( {1 - a^{2}} \right)} - {2\; m\overset{.}{a}}}{2\left( {a + z} \right)^{2}}.}}$

If we choose the function a(t) such that

${\frac{{u\left( {1 - a^{2}} \right)} - {2\; m\overset{.}{a}}}{2\left( {a + z} \right)^{2}} = {\frac{e^{2\; i\;\omega\; t} - 1}{2}u}},$

with the initial condition a(0)=1, namely, a(t) satisfies the Riccatiequation

$\begin{matrix}{{\overset{.}{a} = {{{- \frac{u\;\beta}{2\; m}}a^{2}} - {\frac{{uz}\left( {\beta - 1} \right)}{m}a} + \frac{u\left( {1 + z^{2} - {z^{2}\beta}} \right)}{2\; m}}},{{a(0)} = 1},} & (13)\end{matrix}$

where β=e^(2iωt), then f follows{dot over (f)}=iωf+½uf ²+½βu,  (14)with f(0)=1 since m(0)=1, z(0)=0, and a(0)=1. Now, letg(t)=e^(−iωt)f(t), that is, f in the rotating frame with respect to ω,then we obtainġ=½(g ²+1)ue ^(iωt).  (15)

By the separation of variables, we have

$\begin{matrix}{{{\int_{g_{0}}^{g_{1}}\frac{2\;{dg}}{g^{2} + 1}} = {\int_{0}^{T}{{ue}^{i\;\omega\; t}{dt}}}},} & (16)\end{matrix}$

where g₀=g(0)=1 and g₁=g(T). If the control function satisfies (5), thenwe have

${{\int_{g_{0}}^{g_{1}}\frac{2\;{dg}}{g^{2} + 1}} = {- \frac{\pi}{2}}},$

which, by contour integration, leads to g1=g(T)=0, and hence f(T)=0.More specifically, the contour integration and the residue theorem [3]applied to (16) with the condition (5) gives

$\begin{matrix}{{\int_{1}^{c + {id}}\frac{2\;{dg}}{g^{2} + 1}} = {{2\pi\;{iw}_{1}{{Res}\left( {\frac{2}{g^{2} + 1},i} \right)}} + {2\pi\;{iw}_{2}{{Res}\left( {\frac{2}{g^{2} + 1},{- i}} \right)}} -}} \\{{{\int_{d}^{0}\frac{2\;{idy}}{\left( {c + {iy}} \right)^{2} + 1}} - {\int_{e}^{1}\frac{2\;{dx}}{x^{2} + 1}}},} \\{{= {{2{\pi\left( {w_{1} - w_{2}} \right)}} + {\int_{0}^{d}\frac{2\;{idy}}{\left( {c + {iy}} \right)^{2} + 1}} + {2\;\tan^{- 1}c} - \frac{\pi}{2}}},} \\{{= {{{2{\pi\left( {w_{1} - w_{2}} \right)}} + {2\;\tan^{- 1}\left( {c + {id}} \right)} - \frac{\pi}{2}} = {- \frac{\pi}{2}}}},}\end{matrix}$

where g₁=c+id, c, d∈R, and w₁ and w₂ are the respective winding numbersof the poles i and −i. The above equation is reduced, by the logarithmicform of the arctangent function, to

$\begin{matrix}{{{{\frac{i}{2}{\ln\left\lbrack \frac{c^{2} + \left( {1 + d} \right)^{2}}{c^{2} + \left( {1 - d} \right)^{2}} \right\rbrack}} + {\tan^{- 1}\left\lbrack \frac{2c}{1 - c^{2} - d^{2}} \right\rbrack}} = 0},} & (17)\end{matrix}$

Therefore, we have

${{\tan^{- 1}\left\lbrack \frac{2c}{1 - c^{2} - d^{2}} \right\rbrack} = {{0\mspace{14mu}{and}\mspace{14mu}{\ln\left\lbrack \frac{c^{2} + \left( {1 + d} \right)^{2}}{c^{2} + \left( {1 - d} \right)^{2}} \right\rbrack}} = 0}},$which lead to c=0 and d=0 and give g₁=0.

This result shows that a control that steers the spring modeled in (1)from (π/2, 0)′ to (0, 0)′, which satisfies (5), will drive the functionf from f(0)=1 to f(T)=0. It follows that this steering control is acandidate π/2 pulse for the spin system, because, from (11),

$\quad\begin{matrix}\left\{ \begin{matrix}{{\left( {{m(0)},{z(0)}} \right)^{\prime} = \left( {1,0} \right)^{\prime}},} & {{{{when}\mspace{14mu}{f(0)}} = {{1\mspace{14mu}{with}\mspace{14mu}{a(0)}} = 1}},} \\{{\left( {{m(T)},{z(T)}} \right)^{\prime} = \left( {0,1} \right)^{\prime}},} & {{{{when}\mspace{14mu}{f(T)}} = 0},{{{provided}\mspace{14mu}{a(T)}} < \infty},}\end{matrix} \right. & (18)\end{matrix}$

which correspond to M (0)=(1, 0, 0)′ and M (T)=(0, 0, 1)′, respectively.Note that when m(T)=0, a(t) may have singular solutions at t=T, whichwill be analyzed in Section 2.2. An example minimum-energy controlu*_(π/2)(t)=−cos(3t), as expressed in (7) for ω=3 and T=π, that steersthe spring from X₀=(π/2, 0)′ to X_(F)=(0, 0)′, and the evolutions ofa(t), f(t), and g(t) as described in (13), (14), and (15), respectively,resulting from this optimal control are illustrated in FIG. 4.Similarly, the same concept introduced above can be adopted to design aπ pulse, which is realized by constructing a control that steers thespring between (π, 0)′ and (0, 0)′ or simply by concatenating a π/2pulse with its time-reversed version. The minimum-energy π pulse,u*_(π)=−2 cos(3t), t∈[0, π], and the resulting trajectories for thespring and the spin of ω=3 and T=π are illustrated in FIG. 5.

Section 2.2 Regular Singular Solutions of a(t)

The Riccati equation for a as in (13) can be reduced to a second orderlinear ordinary differential equation of the form [4]Ä−R(t){dot over (A)}+S(t)A=0,  (19)where

$\begin{matrix}{{{R(t)} = {{- \frac{{uz}\left( {\beta - 1} \right)}{m}} + \frac{\left( {\overset{.}{u}\;\beta} \right)}{u\;\beta} - \frac{\overset{.}{m}}{m}}},} & (20) \\{{{S(t)} = {- \frac{u^{2}{\beta\left( {1 + z^{2} - {z^{2}\beta}} \right)}}{4m^{2}}}},} & (21)\end{matrix}$

with the relation

$\begin{matrix}{\frac{\overset{.}{A}}{A} = {a{\frac{u\;\beta}{2m}.}}} & (22)\end{matrix}$

Note that {dot over (A)}(0)=½u(0)A(0) since β(0)=1, a(0)=1, and m(0)=1.In addition to the case m(T)=0 as in (18), the other possibility for

${f(T)} = {\frac{m(T)}{{a(T)} + {z(T)}} = 0}$to hold is when a(T)=∞.

This occurs when the coefficients of (19), i.e., R(t) and S(t) in (20)and (21), respectively, develop singularities at t=T, which in turnoccurs when m(T)=0. Let

$\begin{matrix}{{L_{1} = {\lim\limits_{t\rightarrow T}\left\{ {\left( {T - t} \right){R(t)}} \right\}}},} & (23) \\{L_{2} = {\lim\limits_{t\rightarrow T}{\left\{ {\left( {T - t} \right)^{2}{S(t)}} \right\}.}}} & (24)\end{matrix}$

If both the limits L1 and L2 are finite, then t=T is a regular singularpoint. Note that since R(t) and S(t) are determined by the controlvariable u, one may expect to design a feasible control such that t=T isregular singular when m(T)=0. In this case, we can write (19) as

$\begin{matrix}{{{\overset{¨}{A} - {\frac{P(t)}{t}\overset{.}{A}} + {\frac{Q(t)}{t^{2}}A}} = 0},} & (25)\end{matrix}$

where R(t)=P (t)/t and S(t)=Q(t)/t². Without loss of generality, we mayassume that this regular singular point is transformed to t=0; or,equivalently, we redefine the time axis by letting t=t−T, so now t=0represents t=T. Then, there exists a solution to (25) around the regularsingular point, t=0, of the form (a Frobenius series) [4],

$\begin{matrix}{{{A(t)} = {\sum\limits_{k = 0}^{\infty}{\alpha_{k}t^{k + r}}}},{\alpha_{0} \neq 0},} & (26)\end{matrix}$

where r is a definite (real or complex) constant, and this solution isvalid in some interval 0<|t|<ρ,

ρ>0. Plugging (26) into (25) yields

$\begin{matrix}{{{\sum\limits_{k = 0}^{\infty}{\left( {k + r} \right)\left( {k + r - 1} \right)\alpha_{k}t^{k + r - 2}}} - {\sum\limits_{k = 0}^{\infty}{{P(t)}\left( {k + r} \right)\alpha_{k}t^{k + r - 2}}} + {\sum\limits_{k = 0}^{\infty}{{Q(t)}\alpha_{k}t^{k + r - 2}}}} = 0.} & (27)\end{matrix}$

Taking the series expansions for P(t)=Σ_(i=0) ^(∞)p_(i)t^(i) andQ(t)=Σ_(i=0) ^(∞)q_(i)t^(i) in (27) and collecting the coefficient oft^(k+r−2) gives

$\begin{matrix}{{{\sum\limits_{k = 0}^{\infty}{\left\lbrack {{\left( {k + r} \right)\left( {k + r - 1} \right)} - {{P(t)}\left( {k + r} \right)} + {Q(t)}} \right\rbrack\alpha_{k}t^{k + r - 2}}} = 0},} & (28)\end{matrix}$

which, for k=0, leads to the indicial equation,F(r)=r(r−1)−p ₀ r+q ₀=0,  (29)

since (28) holds for all k=0, 1, 2, . . . , and α₀≠0.

Furthermore, p0 and q0 in (29) can be represented in terms of thecontrol variable and the state of the spin system. We first recall thatm(T)=0, so we have x(T)=0, y(T)=0, and z(T)=1. Now, let's take theTaylor expansions for the state and control functions around the regularsingular point, which is now t=0, to getu=u ₀ +u ₁ t+u ₂ t ²+ . . . ,  (30)x=x ₁ t+x ₂ t ² +x ₃ t ³+ . . . ,  (31)z=1+z ₁ t+z ₂ t ²+ . . . ,  (32)

where x₀=x(T)=0 and z₀=z(T)=1. Also, because, from the Bloch equationsin (8) with v=0, y=ωx, we obtain

${{y(t)} = {{y_{0} + {\int_{0}^{t}{\omega\;{x(\sigma)}d\;\sigma}}} = {{\int_{0}^{t}{{\omega\left( {{x_{1}\sigma} + {x_{2}\sigma^{2}} + \ldots} \right)}d\;\sigma}} = {{\frac{\omega\; x_{1}}{2}t^{2}} + {\frac{\omega\; x_{2}}{3}t^{3}} + \ldots}}}}\;,$

in which y₀=y(T)=0, and hence

$\begin{matrix}{m = {{x + {iy}} = {{x_{1}t} + {\left( {x_{2} + {i\frac{\omega\; x_{1}}{2}}} \right)t^{2}} + {\left( {x_{2} + {i\frac{\omega\; x_{2}}{3}}} \right)t^{3}} + {\ldots\;.}}}} & (33)\end{matrix}$

In addition, integrating the z-component in the Bloch equations, whichobeys ż=−ux, and employing (30) and (31) yields

$z = {1 - {\frac{u_{0}x_{1}}{2}t^{2}} - {\frac{{u_{0}x_{2}} + {u_{1}x_{1}}}{3}t^{3}} - {\frac{{u_{0}x_{3}} + {u_{1}x_{2}} + {u_{2}x_{1}}}{4}t^{4}} - {\ldots\;.}}$Using the definitions in (20) and (21) and the relationsR(t)=P(t)/t=Σ _(i=0) ^(∞) p _(i) t ^(i) /t andS(t)=Q(t)/t ²=Σ_(i=0) ^(∞) q _(i) t ^(i) /t ²

together with the expressions derived in (30), (32), (33), we obtain

$\begin{matrix}{{p_{0} = {\frac{u_{0}\left( {1 - \beta_{0}} \right)}{x_{1}} - 1}},{q_{0} = \frac{u_{0}^{2}{\beta_{0}\left( {\beta_{0} - 2} \right)}}{4x_{1}^{2}}},} & (34)\end{matrix}$

where β=e^(2iωt)=β₀+β₁t+β₂t²+ . . . with β₀=β(T)=e^(2iωT). Therefore,the indicial equation associated with the second-order ordinarydifferential equation (19) is given by

$\begin{matrix}{{F(r)} = {{r^{2} - {\frac{u_{0}\left( {1 - \beta_{0}} \right)}{x_{1}}r} + \frac{u_{0}^{2}{\beta_{0}\left( {\beta_{0} - 2} \right)}}{4\; x_{1}^{2}}} = 0.}} & (35)\end{matrix}$

Section 2.3 Control Synthesis for Regular Singular Solutions

Given the expression in (35), we can design controls (pulses) and choosetheir durations to manipulate the indicial equation and thus the regularsingular solution of (19) around the regular singular point, that is,t=0. The indicial equation in (35) has two roots, given by

$\begin{matrix}{{r_{1} = \frac{u_{0}\left( {2 - \beta_{0}} \right)}{2\; x_{1}}},\mspace{31mu}{r_{2} = {\frac{{- u_{0}}\beta_{0}}{2\; x_{1}}.}}} & (36)\end{matrix}$

Recall that x₁={dot over (x)}(T) and u₀=u(t). We may now show that a(T)is finite, provided x₁≠0 and u₀≠0.

Lemma 1 If u₀=u(T)≠0 and the resulting x-trajectory has a non-vanishedderivative at t=T, i.e., x₁={dot over (x)}(T)≠0, then a(T)<∞.

Proof. Because u₀≠0 and x₁≠0, we have

r₁≠0, r₂≠0, and

${r_{1} - r_{2}} = \frac{u_{0}}{x_{1}}$

from (36). The regular singular solution to the differential equation(19) depends on the relation of r1−r2, which leads to two cases.

Case I: |r1−r2|=Z+. We first examine the situation when r1>r2, i.e.,u0=ζx1, where ζ is a positive integer. Then, the equation (19) has twonontrivial linearly independent solutions of the formA ₁(t)=t ^(r) ¹ h ₁(t)

where h₁(t)=Σ_(i=0) ^(∞)c_(i)t^(i) with c₀≠0, andA ₂(t)=t ^(r) ² h ₂(t)+cA ₁(t)ln t,

where h₂(t)=Σ_(i=0) ^(∞)d_(i)t^(i) with d₀≠0, and c is a constant thatmay or may not be 0. The general solution of (19) isA(t)=k₁A₁(t)+k₂A₂(t), where k₁, k₂∈

Then, we can compute the limit

                                          (37) $\begin{matrix}{{\lim\limits_{t\rightarrow 0}\frac{t{\overset{.}{A}(t)}}{A(t)}} = {\lim\limits_{t\rightarrow 0}\frac{t\left\lbrack {{k_{1}{\overset{.}{A}}_{1}} + {k_{2}{\overset{.}{A}}_{2}}} \right\rbrack}{{k_{1}A_{1}} + {k_{2}A_{2}}}}} \\{= {\lim\limits_{t\rightarrow 0}\frac{\begin{matrix}{{{tk}_{1}\left( {{r_{1}h_{1}t^{r_{1} - 1}} + {{\overset{.}{h}}_{1}t^{r_{1}}}} \right)} + {{tk}_{2}{c\left( {{r_{1}h_{1}t^{r_{1} - 1}} + {{\overset{.}{h}}_{1}t^{r_{1}}}} \right)}}} \\{{\ln\; t} + {k_{2}{ch}_{1}t^{r_{1}}} + {k_{2}r_{2}h_{2}t^{r_{2}}} + {k_{2}{\overset{.}{h}}_{2}t^{r_{2} + 1}}}\end{matrix}}{{k_{1}h_{1}t^{r_{1}}} + {k_{2}{ch}_{1}t^{r_{1}}\ln\; t} + {k_{2}h_{2}t^{r_{2}}}}}} \\{= {\lim\limits_{t\rightarrow 0}\frac{\begin{matrix}{{k_{1}\left( {{r_{1}h_{1}t^{r_{1} - r_{2}}} + {{\overset{.}{h}}_{1}t^{r_{1} + 1 - r_{2}}}} \right)} + {k_{2}{c\left( {{r_{1}h_{1}t^{r_{1} - r_{2}}} + {{\overset{.}{h}}_{1}t^{r_{1} + 1 - r_{2}}}} \right)}}} \\{{\ln\; t} + {k_{2}{ch}_{1}t^{r_{1} - r_{2}}} + {k_{2}r_{2}h_{2}} + {k_{2}{\overset{.}{h}}_{2}t}}\end{matrix}}{{k_{1}h_{1}t^{r_{1} - r_{2}}} + {k_{2}{ch}_{1}t^{r_{1} - r_{2}}\ln\; t} + {k_{2}h_{2}}}}}\end{matrix}$

provided k≠0. By the transformation

${a(t)} = \frac{2\; m\overset{.}{A}}{u\;\beta\; A}$from (22) with (33) and (37), we obtain the limit of a(t),

${{\lim\limits_{t\rightarrow 0}{a(t)}} = {{\lim\limits_{t\rightarrow 0}\frac{2\left( {{x_{1}t} + \ldots}\mspace{14mu} \right)\overset{.}{A}}{\left( {u_{0} + {u_{1}t} + \ldots}\mspace{14mu} \right)\left( {\beta_{0} + {\beta_{1}t} + \ldots}\mspace{14mu} \right)A}} = {\frac{2\; x_{1}r_{2}}{u_{0}\beta_{0}} = {\frac{2\;{x_{1}\left( \frac{{- u_{0}}\beta_{0}}{2\; x_{1}} \right)}}{u_{0}\beta_{0}} = {- 1}}}}},$

where we employed (30) and (31).

If r2>r1, i.e., u0=−ζx1 with ζ∈Z+, then the two linearly independentsolutions of (19) are of the form A₁(t)=t^(r) ² h₁(t) and A₂(t)=t^(r) ¹h₂(t)+cA₁(t) ln t. Similar calculations as in (37) give

$\underset{\_}{{\lim_{t\rightarrow 0}\frac{t{\overset{.}{A}(t)}}{A(t)}} = r_{1}}$and thus

${\lim\limits_{t\rightarrow 0}{a(t)}} = {\frac{2\; x_{1}r_{1}}{u_{0}\beta_{0}} = {\frac{2\;{x_{1}\left( \frac{u_{0}\left( {2 - \beta_{0}} \right)}{2\; x_{1}} \right)}}{u_{0}\beta_{0}} = {\frac{2 - \beta_{0}}{\beta_{0}}.}}}$

Combining the above two cases, we conclude that

${\lim\limits_{t\rightarrow 0}{a(t)}} = \left\{ \begin{matrix}{\frac{2 - \beta_{0}}{\beta_{0}},} & {{{{if}\mspace{14mu}\frac{u_{0}}{x_{1}}} < 0},} \\{{- 1},} & {{{{if}\mspace{14mu}\frac{u_{0}}{x_{1}}} > 0},}\end{matrix} \right.$

where β₀=β(T)=e^(2iωT). Furthermore, let

${\eta = \frac{2 - \beta_{0}}{\beta_{0}}},$then we have |η|=|2−β0|=where β0=β(T)=e2iωT. Furthermore, let η=2−β₀,then we have |n|=|2−β₀|=√{square root over (5−4 cos(2ωT))}. Therefore, ηis bounded by 1≤|η|≤3, and |η|=1 when T=nπ/ω, and |η|=3 whenT=(2n+1)π/(2ω), where n=0, 1, 2, . . . .

Case II: r1−r2≠0 or |r1−r2|≠Z⁺. In this case, the equation (19) has ageneral solution of the form A(t)=k₁A₁(t)+k₂A₂(t), whereA ₁(t)=t ^(r) ¹ h ₁(t),

in which h₁(t)=Σ_(i=0) ^(∞)c_(i)t^(i) with c₀≠0, andA ₂(t)=t ^(r) ² h ₂(t),

in which h₂(t)=Σ_(i=0) ^(∞)d_(i)t^(i) with d₀≠0. Similar calculations asdescribed in Case I lead to the same result, that is,

${\lim\limits_{t\rightarrow 0}{a(t)}} = \left\{ \begin{matrix}{\frac{2 - \beta_{0}}{\beta_{0}},} & {{{{if}\mspace{14mu}\frac{u_{0}}{x_{1}}} < 0},} \\{{- 1},} & {{{{if}\mspace{14mu}\frac{u_{0}}{x_{1}}} > 0},}\end{matrix} \right.$

so that 1≤|lim_(t→0)α(t)|≤3.

To conclude, in this section, we illustrate that if the limits L1 and L2defined in (23) and (24), respectively, exist, then the second-orderdifferential equation A in (19) has a regular singular solution at t=Tand a(T) is finite. This implies that when f(T)=0, it must be m(T)=0 asdesired. Moreover, the regular singular solution depends on the controldesign, because u0 and x1, which results from u0, determine the indicialequation. Ideally, one can design an admissible control input,satisfying the integral condition in (5), such u₀=u(T)≠0, x₁={dot over(x)}(T)≠0, and t=T is a regular singular point. With these conditionsfulfilled, the control function u(t) for t∈[0, T] drives the spin frome1=(1, 0, 0)′ to e3=(0, 0, 1)′, and hence the time-reversed andsign-inverted control function ũ(t)=−u(T−t) for t∈[0, T] is a π/2 pulsethat excites the spin from the equilibrium state e3 to the excited statee₁.

Section 2.4 Asymptotic Exactness

A curious, but important, characteristic of the spin trajectorycorresponding to the minimum energy control pulse—and, therefore, acharacteristic of the mapping between spin and spring—is that theperformance (the closeness of the final z magnetization to 1) approaches1 as the product ωT increases, i.e., as either or both the frequency ofthe Bloch system or the duration of the minimum energy pulse increases(see FIG. 6). The numerical experiments shown in FIG. 6 indicate that asthe value of ωT increases, a(T) asympototically approaches being aregular singular point, which yields finite values of a(T) and forcesthe asymptotic behavior z(T)→1. Therefore, the framework we establish inthis paper quantifies sufficient conditions for asymptotic exactness. Insome contexts our approach may yield arbitrarily exact excitationpulses; in other applications which require restricted durations, ourfindings provide almost exact excitation pulses with a guide for therelationship between performance, duration, and frequency.

Section 2.5 An Alternative Dynamic Mapping

We now discuss an alternative dynamic mapping scenario that permits theuse of two controls in the analysis and provides a different perspectiveon the dynamic connection between the spring and the spin, inparticular, delivering an explanation for the asymptotic behavior of theperformance z(T), through the asymptotic behavior of a(T). If, insteadof using (13), a(t) is chosen such that

${{{\frac{1}{2}\overset{\_}{\alpha}} + \frac{{\overset{\_}{\alpha}\left( {1 - a^{2}} \right)} - {2\; m\overset{.}{a}}}{2\left( {a + z} \right)^{2}}} = 0},$

with the initial condition a(0)=1, namely, a(t) satisfies thedifferential equation

$\begin{matrix}{{\overset{.}{a} = {\frac{\overset{\_}{\alpha}}{2\; m}\left( {{2\;{za}} + z^{2} + 1} \right)}},\mspace{14mu}{{a(0)} = 1},} & (38)\end{matrix}$

then f follows{dot over (f)}=iωf+½αf ²,  (39)

with f(0)=1 since m(0)=1, z(0)=0, and a(0)=1. Similar to the derivationin Section 2.1, let g(t)=e−iωtf (t), then it follows thatġ=½g ² αe ^(iωt),  (40)and, furthermore,

$\begin{matrix}{{{\int_{g_{0}}^{g_{1}}{\frac{2}{g^{2}}{dg}}} = {\int_{0}^{T}{\alpha\; e^{i\;\omega\; t}{dt}}}},} & (41)\end{matrix}$

=g(0)=1 and g1=g(T). One can show that the line integral

$\int_{\gamma}{\frac{2}{g^{2}}{dg}}$is independent of the path γ that starts with g₀=1 and ends at g1=c+idwith c, d∈R. Let c1 and c2 be two non-homotopic loops in C\{0}, and letD1 and D2 denote the regions inside C1 and C2, respectively. Withoutloss of generality, we assume that 0/∈D1 but 0∈D2. Therefore, C1 ishomologous to 0, and the integrand 2/g² is analytic on D1. Hence, wehave

${{\int_{C_{1}}^{\;}{\frac{2}{g^{2}}d\; g}} = 0},$

and, in addition, the Cauchy's integral formula gives

${{{{{\int_{C_{2}}^{\;}{\frac{2}{g^{2}}d\; g}} = {\frac{n\left( {C_{2},0} \right)}{2\pi\; i}\frac{d}{d\; g}}}}_{g = 0}2} = 0},$

where n(C2, 0) denotes the winding number of C2 with respect to 0. As aresult, the integral

${\int_{C}^{\;}{\frac{2}{g^{2}}d\; g}} = 0$over any loop C, and thus it is path-independent.

Now, consider a path γ along the real axis from g0=1 to c and then alignthe imaginary axis from c to g1=c+id, then the line integral

$\begin{matrix}{{\int_{\gamma}^{\;}{\frac{2}{g^{2}}d\; g}} = {{{\int_{1}^{\; c}{\frac{2}{x^{2}}{dx}}} + {\int_{0}^{d}{\frac{2}{\left( {c + {iy}} \right)^{2}}{d({iy})}}}} = {2{\left( {1 - \frac{c}{c^{2} + d^{2}} + {i\frac{d}{c^{2} + d^{2}}}} \right).}}}} & (42)\end{matrix}$

If the control function satisfies (5), then we obtain, using (41) and(42),d=0, 2(c−1)/c=−π/2.  (43)This gives

${g_{1} = \frac{4}{4 + \pi}},{{{and}\mspace{14mu}{f(T)}} = {\frac{4}{4 + \pi}{e^{i\;\omega\; t}.}}}$

If, in addition, the control drives a(T)→−1, then

${{\frac{4}{4 + \pi}e^{i\;\omega\; T}} = {{f(T)} = {\frac{m(T)}{{a(T)} + {z(T)}} = \frac{x_{1} + {iy}_{1}}{z_{1} - 1}}}},$

where

x₁=x(T)∈

, y₁=y(T)∈

, and z₁=z(T)∈

, with z₁ ²=1−x₁ ²−y₁ ². Solving this yields two real solutions

$\begin{matrix}{{{{(i)\mspace{14mu} x_{1}} = 0},{y_{1} = 0},{z_{1} = 1}}{{{({ii})\mspace{14mu} x_{1}} = {- \frac{8\left( {4 + \pi} \right){\cos\left( {\omega\; T} \right)}}{{\pi\left( {8 + \pi} \right)} + 32}}},{y_{1} = {- \frac{8\left( {4 + \pi} \right){\sin\left( {\omega\; T} \right)}}{{\pi\left( {8 + \pi} \right)} + 32}}},{z_{1} = {- \frac{\pi\left( {8 + \pi} \right)}{{\pi\left( {8 + \pi} \right)} + 32}}},}} & (44)\end{matrix}$

and solution (ii) can be omitted since z(T)>0 by the application of thecontrol u that satisfies (5). This can be seen by considering the caseof ω=0, where following the control satisfying (5), the spin is steeredfrom (1, 0, 0)′ to (0, 0, 1)′ at time T. A continuity argument in w maylead to the required conclusion.

It follows that this steering control is a candidate π/2 pulse for thespin system, because, from (11),

$\begin{matrix}\left\{ \begin{matrix}{{\left( {{m(0)},{z(0)}} \right)^{\prime} = \left( {1,0} \right)^{\prime}},} & {{{{when}\mspace{14mu}{f(0)}} = {{1\mspace{14mu}{with}\mspace{14mu}{a(0)}} = 1}},} \\{{\left( {{m(T)},{z(T)}} \right)^{\prime} = \left( {0,1} \right)^{\prime}},} & {{{{when}\mspace{14mu}{f(T)}} = {\frac{4}{4 + \pi}e^{i\;\omega\; T}}},{{{with}\mspace{14mu}{a(T)}} = {- 1}},}\end{matrix} \right. & (45)\end{matrix}$which correspond to M (0)=(1, 0, 0)′ and M (T)=(0, 0, 1)′, respectively.

Numerical calculations show that following the example minimum-energycontrol u*π/2(t)=−cos(3t) expressed in (7) for ω=3 and T=π, the finalstate a(T)→−1 asymptotically with respect to the final time T−the lengthof the pulse—(see FIG. 6), and the spin is steered from (1, 0, 0)′asymptotically to (0, 0, 1)′. The evolutions of a(t), f(t), and g(t) asdescribed in (38), (39), and (40), respectively, resulting from thisoptimal control are illustrated in FIG. 7.

Moreover, we observe empirical evidence (see FIG. 8) that thecorrelation between the spin rotation angle and the initial state of theharmonic oscillator seem to be an exact correspondence.

We have proven that driving the harmonic oscillator from (π/2, 0)′ to(0, 0)′ achieves a π/2=90° rotation of the spin vector. By simulation wesee that driving the spring from (π/4, 0)′ instead corresponds to arotation of the spin vector by π/4=45°. Thus, we conjecture that anytotal rotation γ in the x-z plane) can be reached by satisfying,

${{2\left( \frac{c - 1}{c} \right)} = {\left. {- \gamma}\Rightarrow g_{1} \right. = \frac{2}{2 + \gamma}}},$according to (43), where (γ, 0)′ is the initial state of the spring andstate of the spin.

Section 2.6 Complex Representation of the Spring Driven by Two Controls

In the preceding section, the alternative derivation provided a proofthat allowed both controls of the spin (u and v) to be nonzero, i.e.,α=u+iv was left general. This provides an opportunity to connect thefull Bloch equations of the spin to the case where the spring is drivenby two controls as well. This provides a clearer explanation of thecorrespondence between spin and spring. We consider the system (1) witha second control that directly forces the position term of the harmonicoscillator,

$\begin{matrix}{{\frac{d}{dt}\begin{bmatrix}{\overset{\sim}{x}(t)} \\{\overset{\sim}{y}(t)}\end{bmatrix}} = {{\begin{bmatrix}0 & {- \omega} \\\omega & 0\end{bmatrix}\begin{bmatrix}{\overset{\sim}{x}(t)} \\{\overset{\sim}{y}(t)}\end{bmatrix}} + {{\begin{bmatrix}1 & 0 \\0 & 1\end{bmatrix}\begin{bmatrix}{u(t)} \\{v(t)}\end{bmatrix}}.}}} & (46)\end{matrix}$

A compact expression for (46) is provided by complex notation with{tilde over (m)}(t)={tilde over (x)}(t)+i{tilde over (y)}(t),α(t)=u(t)+iv(t),{dot over ({tilde over (m)})}=iω{tilde over (m)}+α.  (47)

We can develop analogous conditions for the transfer from

${{\overset{\sim}{m}(0)} = {{\frac{\pi}{2}\mspace{14mu}{to}\mspace{14mu}{\overset{\sim}{m}(T)}} = {0\mspace{14mu}{since}}}},\text{}\begin{matrix}{{\overset{\sim}{m}(T)} = {0 = {{e^{i\;\omega\; T}\frac{\pi}{2}} + {\int_{0}^{T}{e^{i\;{\omega{({T - t})}}}{\alpha(t)}{dt}}}}}} & {~~~~~~~~~~~~~~~~~~~~~~~~~~~} \\{{- \frac{\pi}{2}} = {\int_{0}^{T}{{\alpha(t)}e^{{- i}\;\omega\; t}{dt}}}} & {(48)} \\{= {\underset{\underset{= {- \frac{\pi}{2}}}{︸}}{\int_{0}^{T}{\left\lbrack {{{u(t)}{\cos\left( {\omega\; t} \right)}} + {{v(t)}{\sin\left( {\omega\; t} \right)}}} \right\rbrack{dt}}} +}} & {(49)} \\{i{\underset{\underset{= 0}{︸}}{\int_{0}^{T}{\left\lbrack {{{v(t)}{\cos\left( {\omega\; t} \right)}} - {{u(t)}{\sin\left( {\omega\; t} \right)}}} \right\rbrack{dt}}}.}} & \end{matrix}$

We observe that (48) (spring, two controls) differs from (5) (spring,one control) by the sign of the exponential term. More importantly, wenote that (48) differs from (41) (spin, two controls). This means thatin the general two control case, the conditions for spin and spring nolonger coincide without additional criteria. We observe that the singlecontrol case, in which v(t)=0, is a simple way (sufficient condition) tomake these two expressions equivalent. However, it is instructive toidentify the necessary and sufficient conditions for (48) and (41) to beequivalent in the general case when two controls are allowed, i.e.,establish when

$\begin{matrix}{\underset{\underset{{spring}\mspace{14mu}{(48)}}{︸}}{\int_{0}^{T}{{\alpha(t)}e^{{- i}\;\omega\; t}{dt}}} = {\underset{\underset{{spin}\mspace{14mu}{(41)}}{︸}}{\int_{0}^{T}{{\alpha(t)}e^{i\;\omega\; t}{dt}}}.}} & (50)\end{matrix}$

These integral conditions coincide if and only if∫₀ ^(T) u(t)sin(ωt)dt=0,  (51)∫₀ ^(T) v(t)sin(ωt)dt=0,  (52)

Again, in the case of a single control, v(t)≡0 guarantees (52) and thedesign of u(t) to satisfy (4) guarantees (51). In the general case, (51)and (52) identify the additional criteria required to design controlsthat simultaneously drive the spin and spring.

The general minimum energy control that achieves the desiredtransformation using two controls can be computed as

$\begin{matrix}{{{\alpha(t)} = {{e^{{- i}\;\omega\; t}{W^{- 1}\left( {- \frac{\pi}{2}} \right)}} = {{{- \frac{\pi}{2T}}e^{{- i}\;\omega\; t}} = {\underset{\underset{u{(t)}}{︸}}{{- \frac{\pi}{2T}}{\cos\left( {\omega\; t} \right)}} + {i\left( \underset{\underset{v{(t)}}{︸}}{{- \frac{\pi}{2T}}{\sin\left( {\omega\; t} \right)}} \right)}}}}},} & (53)\end{matrix}$

where the controllability grammian is given byW=∫ ₀ ^(T) e ^(−iωt) e ^(iωt) dt=T,

since in the context of complex numbers A′ in the previous definitionbecomes A†, i.e., the conjugate transpose of A. However, this minimumenergy control differs from the expression in (6) because it assumes twocontrols, whereas (6) assumes only one control.

We further observe, without a formal proof, that due to the linearity ofthe spring, we can easily design controls to start from initial statesthat have nonzero imaginary part

$\left( {{e.g.},{{\overset{\sim}{m}(0)} = {\frac{\pi}{2\sqrt{2}} + {i\frac{\pi}{2\sqrt{2}}}}}} \right)$by independently designing u(t) to accomplish the transfer for the realpart, i.e.,

${\overset{\sim}{m}(0)} = \frac{\pi}{2\sqrt{2}}$and v(t) to accomplish the transfer for the imaginary part, i.e.

${\overset{\sim}{m}(0)} = {i{\frac{\pi}{2{\sqrt{2}}^{1}}.}}$

Importantly—and different from the two-control minimum energy control(53)—this independent design of u(t) and v(t) automatically satisfiesboth (51) and (52). Therefore, when designed independently in thismanner, the combined controls also become controls for the spin (e.g.,driving the spin from ¹For a spin magnetization described in sphericalcoordinates with radial distance ρ=1, azimuthal angle θ, and polar angleφ, the spin cartesian coordinates are M=(cos θ sin φ, sin θ sin φ, cosφ)′ (simply transforming from spherical to cartesian coordinates). Hereφ represents the angle that the spin vector must be rotated to bring itto the final state (0, 0, 1)′. The case in the manuscript considers arotation of π/2. At the end of Section 2.5 we discussed the extension todesign a single control that generates an arbitrary flip angle γ alongeither the x or y axes. Now with two controls we can accommodate anyflip angle along any arbitrary direction. The corresponding spring statecaptures the fact that the overall spin rotation is φ, but is sharedbetween the two controls (which is specified by the angle θ): {tildeover (m)}=ϕ cos θ+i ϕ sin θ. Therefore, while φ is the total rotationangle of the spin achieved by both controls applied together, theeffective rotation that each control must generate is φ cos θ for the ucontrol and φ sin θ for the v control.

${M(0)} = \left( {\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}},0} \right)$to M (T)=(0, 0, 1)′). Specifically, a general transfer, i.e., from{tilde over (m)}(0)→{tilde over (m)}(T) can be achieved by designingu(t) and v(t) to satisfy∫₀ ^(T) u(t)cos(ωt)dt={tilde over (m)}(T)cos(ωT)−{tilde over (m)}(0), ∫₀^(T) u(t)sin(ωt)dt=0,  (54)∫₀ ^(T) v(t)sin(ωt)dt=0, ∫₀ ^(T) v(t)cos(ωt)dt=−{tilde over(m)}(T)sin(ωT),  (55)

This approach aligns with the fact that v(t)=0 when the imaginary partof {tilde over (m)}(0) is zero (the case considered in the mainmanuscript and first proof). FIG. 8 depicts an example of such a controland the corresponding state trajectories.

Section 3 Validation of the Dynamic Transformation

Because the dynamic transformation f: [0, T]→C as in (11) may correspondto an invalid spin trajectory, i.e., a complex-valued trajectory,depending on a(t), which in turn depends on the applied control u(t), itis then essential to examine conditions under which f(t) with f(0)=1uniquely determines a Bloch trajectory (x(t), y(t), z(t))′ on the unitsphere for t∈[0, T]. Let f(t)=f1(t)+if2(t) and a(t)=a1(t)+ia2(t), andthen from (11) we have

$\begin{matrix}{{{{f_{1}(t)} + {{if}_{2}(t)}} = \frac{{x(t)} + {i\;{y(t)}}}{{a_{1}(t)} + {i\;{a_{2}(t)}} + {z(t)}}},} & (56)\end{matrix}$for all t∈[0, T]. Since (x, y, z)′ is a Bloch vector, it satisfies, forall t∈[0, T],x ²(t)+y ²(t)+z ²(t)=1.  (57)

Solving (56) and (57) for (x, y, z)′ yields,

$\begin{matrix}{{{x_{1} = {{f_{1}\left( {a_{1} + z_{1}} \right)} - {a_{2}f_{2}}}},{y_{1} = {{a_{2}f_{1}} + {f_{2}\left( {a_{1} + z_{1}} \right)}}},{z_{1} = {\frac{a_{1} - D}{1 + {f}^{2}} - a_{1}}},{or}}{{x_{2} = {{f_{1}\left( {a_{1} + z_{2}} \right)} - {a_{2}f_{2}}}},{y_{2} = {{a_{2}f_{1}} + {f_{2}\left( {a_{1} + z_{2}} \right)}}},{z_{2} = {\frac{a_{1} + D}{1 + {f}^{2}} - a_{1}}},}} & (58)\end{matrix}$

Where D=√{square root over (−a₂ ²|f|⁴+(1−|α|²)|f|²+1)} and where |f|²=f₁²+f₂ ² and |α|²=α₁ ²+α₂ ².

Therefore, it requires that the discriminant is positive i.e.,−a ₂ ² |f| ⁴+(1−|α| ²)|f| ²+1>0,  (59)in order for (xi(t), yi(t), zi(t))′ to be a Bloch trajectory. Employingthe initial conditions f(0)=1 and a(0)=1, i.e., f₁(0)=1, f₂(0)=0,a₁(0)=1, and a₂(0)=0, the corresponding Bloch trajectory can be uniquelydetermined in terms of f(t) and a(t), which is (x₂(t), y₂(t), z₂(t))′for

t∈[0, T] as in (58), because (x₂(0), y₂(0), z₂(0))′=(1, 0, 0)′. Notethat the discriminant in (59) must be strictly positive, because if itis equal to zero, then D=0 implies (x₁, y₁, z₁)′=(x₂, y₂, z₂)′ for allt∈[0, T], which gives an invalid initial condition (½, 0, −½)′.Furthermore, it follows from (59) that the necessary and sufficientcondition for z₂ to be real, and hence (x₂(t), y₂(t), z₂(t))′ a validBloch trajectory, is

$\begin{matrix}{{0 \leq {f}^{2} < \frac{1 - {a}^{2} + \sqrt{\left( {1 - {a}^{2}} \right)^{2} + {4a_{2}^{2}}}}{2a_{2}^{2}}},} & (60)\end{matrix}$

which determines the regime where the dynamic transformation fdetermines a valid spin trajectory with respect to an applied controlfield. Finally, we note that if a(t)≡1 for all t∈[0, T], then D=1, z₁=−1(the south pole), and

${z_{2} = {\frac{1 - {f\overset{\_}{f}}}{1 + {ff}} = \frac{1 - {f}^{2}}{1 + {f}^{2}}}},$which is the Stereographic projection.

To illustrate the importance of the bound on the dynamic projectionf(t), in FIG. 9 we plot the time evolution of D for the minimum-energycontrol and several counterexample controls, which satisfy the integralcondition in (5), but not the bound in (60). Note that all controlssteer the spring from (π/2, 0)′ to (0, 0)′ but only when (60) issatisfied, as in the case of the minimum energy control, does thecontrol also steer the spin from (1, 0, 0)′ to (0, 0, 1)′.

Section 4 Broadband Pulse Design Via Steering Springs

As described in Section 2 and 3, the design of excitation and inversionpulses that manipulate the spin magnetization at a single frequency ωcan be mapped into a problem of steering springs, for which findinganalytical expressions of optimal steering controls is a straightforwardmanner. It follows from this new finding that a control simultaneouslysteering a family of springs over a defined bandwidth between (0, 0)′and (π/2, 0)′ (or (π, 0)′) is a broadband π/2 (or π) pulse,respectively.

Section 4.1 Analytic Broadband Pulse Design

Analytic broadband pulses can be constructed through studying thesteering problem of an ensemble of harmonic oscillators, given by

$\begin{matrix}{{\frac{d}{dt}\begin{bmatrix}{x\left( {t,\omega} \right)} \\{y\left( {t,\omega} \right)}\end{bmatrix}} = {{\begin{bmatrix}0 & {- \omega} \\\omega & 0\end{bmatrix}\begin{bmatrix}{x\left( {t,\omega} \right)} \\{y\left( {t,\omega} \right)}\end{bmatrix}} + {\begin{bmatrix}{u(t)} \\{\upsilon(t)}\end{bmatrix}.}}} & (61)\end{matrix}$

where ω∈[ω1, ω2]⊂R. The minimum-energy ensemble control that drives thisspring ensemble from an initial state X0(ω)=(x(0, ω), y(0, ω))′ to adesired final state XF (ω)=(x(T, ω), y(T, ω))′ at a prescribed time T<∞and minimizes the total energy of the control fields, i.e.,

J=∫₀ ^(T)[û(t)²+v(t)²]dt, was derived in our previous work, and themathematical details can be found in [5]. A brief description isprovided below in this section.

Letp(t,ω)=x(t,ω)+iy(t,ω),

α(t)=u(t)+iv(t), the system (61) is then transformed to a complexsystem, p′(t, ω)=iωp(t, ω)+α(t), with the initial state p(0, ω)=x(0,ω)+iy(0, ω). Without loss of generality, we consider the case in whichthe frequencies are symmetric to the origin, i.e., ω∈[−β, β]. By thevariation of constants formula, we have at time Tp(T, ω)=e ^(iωT) p(0,ω)+∫₀ ^(T) e ^(iω(T-τ))α(τ)dτ,  (62)

for all ω∈[−β, β], and this gives∫₀ ^(T) e ^(−iωτ)α(τ)dτ=e ^(−iωT) p(T,ω)−p(0,ω)≐ξ(ω).  (63)

Let H₁=L₂[0, T] and H₂=L₂[−B, B] be Hilbert spaces over C. Defining thelinear operator L: H₁→H₂ by(Lα)(ω)=∫₀ ^(T) e ^(−iωτ)α(τ)dτ,  (64)it follows from (63) and (64) that(Lα)(ω)=ξ(ω).  (65)

It can be shown that the operator L is compact and that the minimum normsolution of (65), which corresponds to the minimum-energy control α*, isgiven by

$\begin{matrix}{{{\alpha^{*}(t)} = {\int_{- \beta}^{\beta}{e^{i\;\omega\; t}{\sum\limits_{n = 1}^{\infty}{\frac{1}{\lambda_{n}}\ \left\langle {\xi,{\overset{\sim}{\phi}}_{n}} \right\rangle{\overset{\sim}{\phi}}_{n}d\;\omega}}}}},} & (66)\end{matrix}$

where

${\overset{\sim}{\phi}}_{n} = {e^{{- i}\;\omega\frac{T}{2}}\frac{\psi_{n}}{\psi_{n}}}$and λn=2πκn, in which ψn are the prolate spheroidal wave functions(pswf's) and κ_(n)>0 are the associated eigenvalues [6, 7, 8]. Thecontrol α* that steers the spring ensemble from p(0, ω)=π/2 to p(T, ω)=0is a broadband π/2 pulse and that drives the ensemble from p(0, ω)=π top(T, ω)=0 is a broadband π pulse. A truncated optimal control

${{\alpha_{N}(t)} = {\int_{- \beta}^{\beta}{e^{i\;\omega\; t}{\sum\limits_{n = 1}^{N{(ɛ)}}\;{\frac{1}{\lambda_{n}}\left\langle {\xi,{\overset{\sim}{\phi}}_{n}} \right\rangle{\overset{\sim}{\phi}}_{n}\ d\;\omega}}}}},$will steer the ensemble (61) from p(0, ω) to be within an ε-neighborhoodof the desired final state, namely, B_(e)(p_(F)(ω))={h∈

₂: ∥p_(F)(ω)−h(ω)∥₂<∈} at time T, where the size of ε defines thetolerance of the pulse performance.

Section 4.2 Numerical Synthesis of Broadband Pulses

Note that the pswf's can be approximated by the discrete prolatespheroidal sequences (dpss's), denoted {v_(t,k)(N,W)} which are definedvia the solution to the following algebraic equation [7, 9]

${{\sum\limits_{t^{\prime} = 0}^{N - 1}{\frac{\sin\left\lbrack {2\pi\;{W\left( {t - t^{\prime}} \right)}} \right\rbrack}{\pi\left( {t - t^{\prime}} \right)}{\upsilon_{t^{\prime},k}\left( {N,W} \right)}}} = {{\lambda_{k}\left( {N,W} \right)}{\upsilon_{t,k}\left( {N,W} \right)}}},$

where 0<W<½ and t=0, 1, . . . , N−1. The minimum-energy ensemble controlα* as in (66) and its truncation αN (t) can be easily calculatednumerically using the discrete prolate spheroidal sequences in mostscientific programming tools, e.g., the MATLAB command “dpss”. Abroadband π pulse that produces uniform excitation over the designedbandwidth is shown in FIG. 10. In addition, this optimal control canalso be effectively obtained via finding the minimum-norm solution tothe integral equation (64) using the recently developed SVD-basedalgorithm [10].

Section 4.2.1 Design of Constrained Broadband Pulses

In practice, the amplitude or power of RF pulses may be limited. Suchconstrained broadband pulse design can be formulated as a minimizationproblem, where we wish to steer the spring ensemble p(0, ω)=π/2 or p(0,ω)=π to p_(F)(ω)=0, given by

$\begin{matrix}{{{\min\limits_{u,\upsilon}{\int_{- B}^{B}{{{{p\left( {T,\omega} \right)} - {p_{F}(\omega)}}}^{2}d\;\omega}}},}\ } & (67) \\{{{{s.t.\mspace{14mu}{u^{2}(t)}} + {\upsilon^{2}(t)}} \leq A_{\max}^{2}},} & (68)\end{matrix}$

where A_(max) is the maximum allowable amplitude. Note that p(T, ω)depends on the control α=u+iv as defined in (62). When v=0, this optimalcontrol problem, following some algebraic manipulations, can be reducedto a convex optimization problem of the form

$\begin{matrix}{{\min\limits_{u}{\int_{0}^{T}{\left\lbrack {{\int_{0}^{T}{\frac{\sin\left\lbrack {\beta\left( {\tau - \sigma} \right)} \right\rbrack}{\tau - \sigma}{u(\tau)}{u(\sigma)}d\;\sigma}} + {\frac{2{\sin\left( {\beta\;\tau} \right)}}{\tau}{u(\tau)}}} \right\rbrack d\;\tau}}}{{s.t.\mspace{14mu}{u^{2}(t)}} \leq {A_{\max}^{2}.}}} & (69)\end{matrix}$

Discretizing this convex problem leads to a finite-dimensional quadraticprogram

$\begin{matrix}{{\min\limits_{U}{U^{\prime}{HU}}} + {2U^{\prime}Q}} & (70) \\{{{s.t.\mspace{14mu}{u_{i}}} \leq 1},{i = 1},\ldots\mspace{14mu},n,} & (71)\end{matrix}$

where U=(u1, . . . , un)′, t1=0, tn=T,

${H = \begin{bmatrix}{\sin\;{c\left( {t_{1} - t_{1}} \right)}} & {\sin\;{c\left( {t_{1} - t_{2}} \right)}} & \ldots & {\sin\;{c\left( {t_{1} - t_{n}} \right)}} \\{\sin\;{c\left( {t_{2} - t_{1}} \right)}} & {\sin\;{c\left( {t_{2} - t_{2}} \right)}} & \ldots & {\sin\;{c\left( {t_{2} - t_{n}} \right)}} \\\vdots & \vdots & \ddots & \vdots \\{\sin\;{c\left( {t_{n} - t_{1}} \right)}} & {\sin\;{c\left( {t_{n} - t_{2}} \right)}} & \ldots & {\sin\;{c\left( {t_{n} - t_{n}} \right)}}\end{bmatrix}},$

and Q=(sinc(t1), . . . , sinc(tn))′ in which sinc(x)=sin(x)/x. Theresulting quadratic program can be effectively solved, for example, withstandard gradient methods or using commercial nonlinear programmingsolvers, and its global optima are in a bang-bang form, which hasswitching characteristics between the positive and negative maximumallowable amplitude as illustrated in FIG. 3.

REFERENCES

-   [1] R. W. Brockett. Finite Dimensional Linear Systems. John Wiley    and Sons, Inc., 1970.-   [2] J. Cavanagh, W. J. Fairbrother, A. G. Palmer, and N. J. Skelton.    Protein NMR Spectroscopy. Academic Press, San Diego, Calif., 1996.-   [3] L. Ahifors. Complex Analysis. McGraw Hill, 1979.-   [4] E. L. Ince. Ordinary Differential Equations. Dover Publications,    New York, 1956.-   [5] J.-S Li. Ensemble control of finite-dimensional time-varying    linear system. IEEE Transactions on Automatic Control,    56(2):345-357, 2005.-   [6] C. Flammer. Spheroidal Wave Functions. Stanford University    Press, Stanford, Calif., 1957.-   [7] D. B. Percival. Spectral Analysis for Physical Applications.    Cambridge University Press, Cambridge, UK, 1993.-   [8] D. Slepian and H. O. Pollak. Prolate spheroidal wave function,    fourier analysis and uncertainly—i. Bell System Tech. J., 40:43-64,    1961.-   [9] D. Slepian and H. O. Pollak. Prolate spheroidal wave function,    fourier analysis and uncertainly—v. Bell System Tech. J.,    57:1371-1430, 1978.-   [10] A. Ziotnik and J.-S. Li. Synthesis of optimal ensemble controls    for linear systems using the singular value decomposition. In 2012    American Control conference, Montreal, June 2012.

What is claimed is:
 1. A method of performing broadband excitation orinversion of a two-level spin system, comprising: providing a two-levelsystem comprising at least one qubit (spin) and at least one harmonicoscillator (spring); defining a radio frequency (RF) bandwidth for thetwo-level system; bounding a radio frequency (RF) amplitude or totalenergy for an RF pulse; determining a desired terminal magnetizationprofile or flip angle for the at least one spin; determining a desiredterminal magnetization profile or flip angle for the at least onespring; mapping an endpoint of a trajectory of the spins to an endpointof a trajectory of the springs, wherein the endpoints of the spins andsprings correspond to the desired terminal magnetization profile or flipangle; employing a single control or two controls simultaneously in bothan x direction and a y direction; calculating a converging solution foran RF pulse; or steering the spring and the spin to a desired terminalmagnetization profile or flip angle by applying the calculatedconverging solution for the RF pulse to the two-level system.
 2. Themethod of claim 1, wherein the calculating a converging solutioncomprises generating RF pulse parameters, providing a first controldesign comprising a minimum-energy broadband pulse, or providing asecond control design comprising an amplitude-limited broadband pulse.3. The method of claim 1, wherein (i) the two-level spin system isselected from the group consisting of a logical qubit spin system, anuclear spin system, a photon spin system, an electron spin system, anatomic spin system, and a dot spin system; (ii) a dynamic connectionbetween nonlinear spin and linear spring systems are calculated underoptimal forcing to design an RF pulse based on the design of a controlto steer linear harmonic oscillators; (iii) a condition of a one-to-onecorrespondence to a spin trajectory is satisfied; (iv) the RF pulseconditions result in the spins and springs having coincidingmagnetization on the same axis and excitation or inversion of at least99% of the spins and springs; (v) the RF pulse compensates for adistribution of spin and spring frequencies; (vi) the RF pulse is anexcitation pulse, a reverse excitation pulse; or an inversion pulse; or(vii) the RF pulse results in a spin flip angle selected from π, π/2, orπ/4.
 4. The method of claim 1, wherein the RF pulse is computed usinglinear systems to force the spring.
 5. The method of claim 1, whereinthe RF pulse is an excitation pulse or an inversion pulse for anonlinear Bloch system.
 6. A method of constructing an RF pulsecomprising: (i) obtaining an energy parameter; (ii) obtaining abandwidth parameter; (iii) obtaining a desired flip angle (magnetizationprofile), wherein the flip angle is between 0 and 180 degrees; (iv)converting the desired flip angle from a spatial coordinate system to alinear coordinate system; (iv) inputting the energy parameter, thebandwidth parameter, and the linear coordinates into the system; or (v)analytically deriving parameters for the RF pulse, wherein the RF pulseproduces a desired magnetization profile of a final spin or springstate.
 7. The method of claim 6, wherein the bandwidth parametercorresponds to a range of frequencies of a sample; the method can beperformed on-line; or the method uses analytical approaches oroptimization-free algorithms.
 8. The method of claim 6, comprising:providing a two-level system comprising at least one qubit (spin) and atleast one harmonic oscillator (spring) and mapping the spins to thesprings; providing a first control design comprising a minimum-energybroadband pulse; or providing a second control design comprising anamplitude-limited broadband pulse.
 9. The method of claim 6, wherein theRF pulse (i) produces a desired distribution of final spin or springstates or a desired magnetization profile when applied to a two-levelspin system; (ii) achieves at least 99% broadband excitation orinversion; and (iii) is not a hyperbolic secant pulse.
 10. The method ofclaim 6, wherein the RF pulse (i) compensates for a distribution of spinand spring frequencies; or (ii) satisfies experimental requirementsselected from a bound on an RF pulse amplitude or total energy of the RFpulse.
 11. The method of claim 6, wherein the bandwidth parameter isbetween −40 kHz and 40 kHz.
 12. The method of claim 6, wherein theenergy parameter comprises a maximum allowable power in Watts or dbm.13. The method of claim 6, wherein flip angle is between 0 and 360° orselected from a π, a π/2, or a π/4 flip angle.
 14. A method of designinga broadband radiofrequency (RF) pulse comprising: (i) providing atwo-level spin system comprising at least one qubit (spin) and at leastone harmonic oscillator (spring); (i) employing at least a first controldesign and optionally, a second control design simultaneously in both anx and a y direction; or (ii) achieving a desired flip angle of a qubit,wherein the flip angle of a qubit is controlled by steering the springbetween specific states.
 15. The method of claim 14, wherein the qubitis selected from the group consisting of a logical qubit, a nuclearspin, a photon, an electron, an atomic spin, and a dot spin.
 16. Themethod of claim 14, wherein the qubit is a nuclear spin and a broadbandRF pulse compensates for a distribution of spin and spring frequencies.17. The method of claim 14, further comprising numerical optimization,resulting in an RF pulse or an RF pulse sequence that places bounds on aradio-frequency (RF) amplitude or a total energy of the RF pulse. 18.The method of claim 14, wherein the first control design comprises aminimum-energy broadband pulse; and the second control design comprisesan amplitude-limited broadband pulse; the flip angle can be betweenabout 0° and 360°; or the RF pulse performs at least 99%, exact, orasymptotically exact excitation or inversion over a defined bandwidth,optionally with a bounded amplitude.
 19. The method of claim 14, whereinthe RF pulse has a bang-bang pulse shape.
 20. The method of claim 14,wherein an excitation or inversion of the spins can be adjusted byselecting different amplitude bounds and RF pulse durations.